We should start by understanding what the deflection of the "inclinometer" ball actually indicates. To a good approximation, it responds to the net sum of all lateral (sideways) accelerations acting on an aircraft, excluding the lateral acceleration component due to gravity. By "lateral" or "sideways" we mean in the aircraft's own reference frame -- for example, in a turn, the direction meant by "lateral" or "sideways" would be constantly changing as seen by an observer on the ground, or for that matter, an observer in a balloon drifting by with the airmass. In a normal coordinated turn, the net lateral acceleration other than that due to gravity is zero. To a first approximation, the only thing that causes an aircraft to experience a lateral acceleration, other than gravity, is the aerodynamic sideforce generated by a sideslip as the airflow strikes the side of the fuselage. That is why the ball serves as a slip-skid indicator, i.e. a sideslip indicator. There are some reasons why the ball is not a perfect indicator of sideslip, which will become evident later in this answer, but the idea that the ball is basically a sideslip indicator is a good starting point.
Although it's not germane to this answer, to be complete we should also point out that for a given aerodynamic sideforce, the motion of the ball is also influenced by the G-loading, or more precisely, by the upward aerodynamic force generated by the wing. The higher the "upward" G-loading, the more aerodynamic sideforce it takes to cause a given deflection of the ball. At zero "upward" G's, the slightest aerodynamic sideforce will put the ball all the way into one of the far corners of the glass tube, and at negative "upward" G's, the ball tends to just stay stuck in one of the corners even when sideforce is zero. This is why many aerobatic aircraft have a second slip-skid ball that is mounted upside-down.
If the ball is an accelerometer, how can it be deflected in a situation like a steady-state wing-down slip along a straight-line flight path, where net acceleration is clearly zero? The answer is that net acceleration is zero, but there is still an aerodynamic force component, and therefore an aerodynamically-driven acceleration component, acting in the sideways direction in the aircraft's reference frame, caused by the air impacting the side of the fuselage. Just as a panel-mounted G-meter reads "1" not "0" in straight-and-level flight-- net acceleration is zero, but there is still an aerodynamic force component acting in the upwards direction in the aircraft's reference frame-- the lift force from the wing. Just as the slip-skid ball measures the net lateral (sideways) acceleration other than that caused by gravity, so too does the G-meter measure the net vertical acceleration component other than that caused by gravity, which is essentially just the upward acceleration caused by the wing's lift vector.
What reference frame is the slip-skid ball operating in? We've seen that the slip-skid ball measures lateral acceleration (other than that due to gravity). We can measure net acceleration in any valid inertial reference frame and we'll get the same answer, except for differences due to the tilt of one frame relative to the other. An accelerating airmass is not a valid inertial reference frame. Nor is the aircraft itself, unless net acceleration is zero. Therefore it is not wrong to assert that in the context of an accelerating airmass, the slip-skid ball is operating in the reference frame of the earth, rather than the reference frame of the aircraft or the airmass, except that the reference frame is tilted to match the orientation of the aircraft in space at any given instant in time, and so is constantly changing its orientation with respect to the orientation of the earth's reference frame. In other words the direction called "lateral" is fixed with respect to the aircraft, not the earth-- it changes (relative to the earth) as the aircraft changes heading, or as the aircraft banks. Yet the actual acceleration in that (ever-changing) lateral direction is measured in relation to a valid inertial reference frame, not in relation to the aircraft itself. It is a valid question to ask how it can be that the slip-skid ball which in a sense is tied to the earth reference frame, can give essentially the same indication as the yaw string. Perhaps the above paragraphs have shed some light on those questions-- the answer has to do with the fact that the slip-skid ball is only measuring the lateral component of acceleration, and the fact that the lateral component of acceleration is intimately related to sideslip. Keep in mind that in many instances in this answer, we use phrases like "in the aircraft's own reference frame", but we really only mean to describe the orientation of the reference frame involved. At any instant time, the actual reference frame that an on-board accelerometer such as the G-meter or the slip-skid ball is operating in would indeed be an actual valid inertial reference frame such as that of the earth itself, just tilted differently. It is correct to observe that an accelerometer is not actually operating in the reference frame of the aircraft itself in any case where the aircraft is accelerating, or in the reference frame of the airmass in any case where the wind is not steady. If the slip-skid ball were actually operating in the reference frame of the aircraft itself, it would always read zero, and likewise the G-meter.
Your first specific question is essentially "if the pilot is keeping the yaw string centered, will the presence of a steady crosswind cause the ball to deflect?" Absolutely not. The plane flies within the airmass, or "inside of the wind". It doesn't "feel" a steady wind in any way. Like if you were flying a tiny model airplane inside a car driving steadily down the highway at 100 mph-- the plane doesn't know which way the car is driving. The pilot doesn't have to do anything different with the rudder to keep the yaw string (or ball) centered when there is a steady crosswind, than when there is not.
You'll probably have an easier time understanding the effects of crosswind gusts if you assume that the pilot is not using the rudder pedals to try to keep the yaw string perfectly centered. The gust will deflect the yaw string downwind and the ball upwind until the aircraft has accelerated sideways sufficiently to be back in equilibrium with the new state of movement of the airmass. During that time of acceleration, while the airplane is "feeling" the sideways wind component, the vertical fin is also exerting a yaw torque to yaw the nose into the gust. How much change in heading happens before the aircraft reaches equilibrium with the new state of the airmass will depend on how big the fin is and how much rotational inertia the aircraft has about the yaw axis. This will vary from one aircraft to another. Of course, the change in heading causes the acceleration from the gust to become less of a lateral acceleration and more of a longitudinal acceleration, in the aircraft's reference frame.
Lest anyone get the wrong idea, we should note that a pilot usually does not apply rudder to correct for a crosswind gust, except perhaps during final approach when the wheels are about to touch the ground and small deviations in the aircraft heading cannot be tolerated-- and then the correction is in the opposite direction that we are talking about here. As far as keeping the nose pointing into the relative wind, normally the vertical tail takes care of things well enough on its own, by creating a rudder-like effect that yaws the aircraft. This is sometimes called the "weathervane effect", although it is important to understand to understand that it is driven only by the "relative wind" that is felt by the aircraft, and is not influenced in any way be a steady wind. Since this yawing does not happen instantly, the side of the aircraft is temporarily exposed to the sideways airflow component created by the gust, so we'll see a temporary deflection of the ball in the upwind direction and the yaw string in the downwind direction.
A recent addition to the question has raised the question of what behaviour we would see in an aircraft with essentially zero mass. How will such an aircraft respond in the presence of a sideways gust, or a steady increase in horizontal wind speed? The aircraft will accelerate instantly with the gust or the increase in wind speed and there will be no tendency at all for the vertical fin to generate any yaw torque at all. The aircraft heading will have no tendency to change. Yet the yaw string will stay centered. Because there is lateral acceleration, the slip-skid ball will indeed not stay centered -- assuming that the slip-skid ball somehow does exist in such an aircraft. What we've done here is by specifying zero mass, we've broken the intimate connection between sideslip and lateral acceleration. With an infinitely small amount of mass, an infinitely small amount of sideslip is needed to drive a given lateral acceleration. In the real world, we will indeed see less deflection of the slip-skid ball for a given deflection of the yaw string when a gust strikes a heavily loaded aircraft, than when it strikes an aircraft of identical shape and size but less mass. A much simpler analogy would be two toy cars of the same shape and size but different mass, each with a flag on it. When a wind gust strikes both cars and blows both flags in an identical manner, the wind will accelerate the lighter car faster than the heavier car. The flags are analogous to the yaw string, and the cars are analogous to the aircraft accelerating sideways and tending to "leave the ball behind", so that the ball deflects sideways inside its tube.
Now for the part of your question about the superman pilot with lightning-fast reflexes making control inputs as needed-- if needed-- to keep the yaw string perfectly centered as a wind gust strikes. In a real aircraft, with non-zero mass and a non-zero moment of rotational inertia about the yaw axis, some control inputs will definitely be needed -- in essence, the pilot will be doing some fancy footwork on the rudder pedals to "help" the vertical fin do a better job of keeping the nose of the plane pointing directly into the "relative wind" so that the yaw string stays absolutely perfectly centered even as a strong gust strikes from the side. Some aileron input may be needed as well to keep the wings level-- surely we don't want to complicate the problem by allowing the plane to bank as the gust strikes.
Let's start by simplifying the problem by assuming that the rudder pedals are connected to wingtip drag-producing devices that, unlike a conventional rudder, generate no net sideforce when they are deployed into the airflow. They only generate yaw torque.
Basically we're saying that the pilot is using the rudder pedals to "help" the fin "weathervane" the nose to point directly into the relative wind, in such a perfect manner that the plane never experiences any sideslip at all even as a gust strikes.
This means that the aircraft will never feel lateral accelerations, only longitudinal accelerations. As the gust strikes, the aircraft will tend to pitch up and climb, but to a first approximation the ball will remain centered along with the yaw string.
Why only "to a first approximation"? Well, the ball is not quite perfect as a sideways accelerometer. The rate of yaw rotation about the aircraft's CG can also influence the ball. Imagine that we mounted the aircraft on a pivot at its CG in a vacuum-filled hangar and twirled the airplane like a pinwheel in the yaw dimension. The ball would deflect opposite the direction that the nose was moving. The further the cockpit from the CG, the more pronounced this effect would be.
Normally this effect is negligible in actual flight. How important it is in your thought experiment depends on how "sharp-edged" the gust is-- how quickly it ramps up. Does it take two seconds to reach full strength? Half a second? Zero time at all? If the latter, the aircraft must yaw at an infinite rate of speed to stay aligned with the relative wind and keep the yaw string centered, and the ball will be ejected out the downwind side of the tube!
In a later edit you suggested that you were interested in the case where the wind just keeps getting stronger and stronger indefinitely. Let's continue with our "fancy footwork" thought experiment where the pilot "helps" the fin to eliminate all sideslip so that the yaw string stays centered, and (to a first approximation) the ball stays centered too. Again, in the aircraft's own reference frame, no lateral (sideways) acceleration is taking place, only longitudinal acceleration. If the wind is initially zero, and starts blowing directly cross to the aircraft's original course, then as the wind gets stronger and stronger, before long the aircraft will be pointing 45 degrees off its original heading, and then before much longer the aircraft will be pointing essentially directly into the wind, and there will be no need for any more rudder pedal inputs to keep the yaw string and ball centered, even as the wind speed continues to increase. Of course, the aircraft's groundspeed will soon be negative, and getting more and more so. While the aircraft will have maintained its original groundspeed component in the direction that is cross to the wind, this will soon have a negligble effect on the direction of the ground track because the groundspeed component in the downwind direction will be so large. If the elevator and throttle are left in the same position that originally yielded level flight, the airspeed will stabilize at a value that is close to its original value, but the aircraft will be climbing, due to the extra energy constantly added into the system by the constant increase in wind speed. The situation has parallels to the "dynamic soaring" method used by the albatross to exploit the wind gradient over the open ocean.
At this point you may feel the plot has gotten convoluted enough and the story should end. And it probably should. If you are still having any trouble understanding any of the above content, please stop reading now and review! Especially if you still are not convinced that an aircraft does not "feel" the presence of a steady wind. Your time and effort are better spent trying to understand that concept, then in diving into any of the micro-level details that follow.
But if you feel ready for yet a few more twists and turns--
What happens if the pilot uses a rudder to yaw the aircraft? A rudder generates its own aerodynamic sideforce, even when the rest of the aircraft does not. When we step on the left rudder pedal, we deflect the rudder to the left, creating an aerodynamic sideforce to the right as well as a yaw torque to the left. Often this yaw torque changes the orientation of the aircraft relative to the flight path, so that the right side of the aircraft is exposed to the airflow, which creates an aerodynamic sideforce to the left that dwarfs the opposing sideforce from the rudder itself. (See for example this related ASE answer.) So the ball is deflected to the right-- opposite the direction that the yaw string is deflected. But in the case of our thought experiment with the wind gust and the "perfect" correction by the pilot, the aircraft stays perfectly streamlined to the airflow, and nothing opposes the rightwards aerodynamic sideforce from the deflected rudder, so the ball is deflected to the left-- in the same direction that the pilot deflected the rudder. This comes into play with dealing with a failed engine in a twin-engined aircraft-- see footnote 1 for more.
Again, how important this effect is in our thought experiment depends on how "sharp-edged" the gust is-- how quickly it ramps up. Does it take two seconds to reach full strength? Half a second? Zero time at all? If the latter, the aircraft must yaw at an infinite rate of speed to stay aligned with the relative wind and keep the yaw string centered, and the sideforce from the deflected rudder would tend to cause the ball to be ejected out the "upwind" side of the tube!
And now the million-dollar question-- in a gust that ramps up at a non-infinite rate, which of the two above effects dominates? The centrifugal force from the rotation of the aircraft about its own CG, which tends to move the ball in the "downwind" direction, or the sideforce from the deflected rudder, which tends to move the ball in the "upwind" direction? Almost certainly the latter, for most real aircraft. But let's consider the extreme cases--
Extreme case 1-- the rudder is very far aft of the CG, and the aircraft has a very low moment of rotational inertia about the yaw axis, and very little aerodynamic damping in yaw (e.g. a small vertical fin) so that only a very little sideforce from the rudder is needed to establish a given rate of rotational acceleration in the yaw axis, or to maintain a given rate of yaw rotation. The cockpit is very far in front of the CG. The first effect-- the "centrifugal force" effect-- will dominate and the ball will move in the opposite direction as the rudder input.
Extreme case 2-- The rudder is very large but is only a very short distance behind the CG, and the aircraft has a very high moment of rotational inertia about the yaw axis, and lots of yaw damping (e.g. lots of side area behind the CG), so that lots of sideforce from the rudder is needed to establish a given rate of rotational acceleration about the yaw axis, or to maintain a given rate of yaw rotation. The cockpit is only a short distance in front of the CG. The second effect-- the sideforce from the rudder-- will dominate and the ball will move in the same direction as the rudder input.
Now we've gone and opened another can of worms here as well-- the difference between the rudder input required to establish a given rate of rotational acceleration about the yaw axis, versus the rudder input required to maintain a given rate of rotation about the yaw axis. What is the appropriate response to a gust that ramps up linearly? Or does the gust perhaps increase in a sinusoidal manner rather than a linear one?
Note that to keep a steady yaw rotation rate going once established, we'll still need to keep the rudder deflected. This is due to "yaw damping"-- if the nose of the aircraft (where the yaw string is located) is perfectly streamlined to the airflow, then the further-aft parts of the aircraft cannot be, due to the differences in the velocity of the local relative wind induced by the yaw rotation itself. (E.g., points on the tips of different blades of a pinwheel or windmill are moving through the airmass in different directions at any instant in time.) And where is the torque that creates this yaw damping coming from? Actually, in this case where the nose of the aircraft is specified to be streamlined to the flow, it's coming largely from the rear parts of the aircraft being pushed sideways through the air. So we do have some aerodynamic sideforce opposing the sideforce from the rudder, after all. If the rudder is deflected and the yaw rotation rate is constant and there is no opposing yaw torque (like that due to a failed engine-- we're assuming the increased drag from the faster-moving outboard wingtip is negligible in the grand scheme of things), then we have to conclude that the net sideforce is in fact in the same direction as the rudder is deflected, after all. So in this case, after our initial yaw rotation rate is established, the ball will be deflected in the opposite direction as the rudder is deflected. Both the "centrifugal" effect from the yaw rotation, and the effect from sideforce, are working together. (And to be complete, we need to point out that this sideforce from yaw damping also exists in the case of the aircraft that is yawed with tip-dragger devices.) But when we were first establishing the yaw rotation rate, the ball may have been doing something different.
Don't spend any time trying to understand these nuances until you are first completely solid on the idea that an aircraft does not feel the presence of a steady wind in any way. In aviation, that's kindergarten stuff. To complete the analogy, getting into the nuances of differences between the behavior of the yaw string and the ball in various situations involving rudder deflections and yaw rotations is more of a graduate-school-level topic. Understanding that the ball responds mainly to aerodynamic sideforce, and the rudder creates some sideforce whenever it is deflected even if the side of the fuselage is not exposed to the airflow-- that falls somewhere in between (high-school level?), and pilots routinely put that into practice when they leave the ball deflected toward the working engine when practicing for engine failure, or dealing with actual engine failure in a twin-engined airplane.
Postscript 1-- I suspect that what you really want to ask, is "what happens if the aircraft magically rotates as needed to stay perfectly aligned with the relative wind as the crosswind gust ramps up, while also bending like a banana so that the yaw rotation itself creates no variations in the direction of the local relative wind experienced by various parts of the aircraft." In this case the ball moves in the "downwind" direction to some degree that is determined by the aircraft's yaw rotation rate, due to the centrifugal force effect, but the aircraft experiences no lateral acceleration, only longitudinal acceleration.
Footnote 1-- This is like what happens when we use the rudder to control a twin-engine aircraft with one failed engine-- when the fuselage is streamlined to the airflow, the ball is slightly deflected in the direction of the deflected rudder, so we should refrain from applying as much rudder as would be needed to fully center the ball and bring the turn rate to zero with the wings level. Instead we should leave the ball slightly deflected in the same direction as our rudder input (i.e. toward the good engine), and stop the turning tendency -- which fundamentally is caused by the sideforce from the deflected rudder -- by banking slightly toward the good engine.