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I have read and seen some articles in which, for various reasons, the aircraft's nose gear collapsed after landing during the roll out. This resulted in the aircraft going off the runway.
Is it possible to still control the aircraft during this weird event? Or is it more or less inevitable that the aircraft will run off the runway?
Most aircraft are steered on the ground by pivoting the nose gear. If the nose gear collapses, you therefore lose your primary method of directional control on the ground.
Additionally, parts of the aircraft contacting the ground that are not supposed to contact the ground can cause unwanted yaw, which can contribute greatly to the runway excursions you've noticed in articles regarding gear collapse incidents. This is probably the largest contributing factor to runway excursions in cases of nose gear collapse. However, if it's known that the nose gear did not deploy properly before the landing, the pilots can intentionally hold the nose gear off the ground as long as possible in order to minimize this effect. Holding the nose gear off the ground as long as possible minimizes the time that the contact between the nose and runway surface will be potentially applying yawing forces to the aircraft and also minimizes the speed of the aircraft while those forces are being applied. This gives the pilots more time to respond with differential braking and/or rudder, helping to prevent the nose from moving too far away from the direction of travel and a resulting loss of directional control due to skidding. Of course, this doesn't help if the gear collapse was unexpected, as the pilots wouldn't know to hold the nose gear up.
There are a few additional methods of directional control available to airplanes on the ground, though most are not as effective as pivoting the nose gear and their effectiveness varies with speed:
Rudder - While the aircraft still has high airspeed when rolling down the runway, the rudder will still provide some directional control. This control will diminish very rapidly as the aircraft slows down, though.
Asymmetric braking - Most airplanes allow the pilots to control the brakes on the left and right main gear independently, allowing them to brake harder on one side of the aircraft than on the other. This is more useful for taxi speeds than for high speeds down a runway, though, and can easily cause even more directional control problems if used without proper care at high speeds.
Asymmetric thrust - Multi-engine aircraft almost always allow the pilots to control the thrust (either forward or reverse) of each of the engines independently. However, in the case of a collapsed gear, this is unlikely to be very helpful, due to the large delay between when a thrust change is commanded and when it happens as well as that the engines may now also be striking the ground themselves.
There is nothing wrong with all answers here, but one aspect is missing.
If the nose gear collapses, the instinctive reaction of most pilots would be to brake as much as possible. Forget reverse thrust or spoilers - the pilot will hit the brakes. If pressure builds up slightly differently between left and right brake, or if the wheel load between both main landing gears is different due to cross wind, the aircraft will experience asymmetric braking, and now the nose will not be exactly centered between both main wheels.
Now we need to look at the braking forces of the skidding front fuselage in comparison to those of the main wheels. Since the aircraft has now a pronounced nose-down attitude, its load on the nose will be quite a bit higher than the regular load on the nose landing gear used to be. With the increased load on the front fuselage, its scratching along the concrete will contribute a lot to the braking force, and if this force is off-center it will produce a yawing moment that can be balanced only initially and with full application of differential braking. Once the front fuselage has sverved beyond the tracks of the main gear, no braking will bring it back into the center and the aircraft is uncontrollable. Add some crosswind and it will leave the runway quickly.
Is it possible to still control the aircraft during this weird event?
Applying rudder will still help at high speed, but once the aircraft slows down and the rudder becomes less effective, the aircraft is uncontrollable.
Or is it more or less inevitable that the aircraft will run off the runway?
Only in perfectly symmetrical conditions will it stay on the runway. Since the aircraft with the collapsed nose gear is unstable in yaw, the far more likely outcome is a runway excursion.
To an extent, yes.
Most importantly you can also control the airplane using differential braking of the rear wheels. This is how planes with a free castering nose wheel are steered on the ground.
On a side note if the plane is moving fast enough and their is sufficient airflow over the rudder you may be afforded limited control of the aircraft via the rudder. However as the speed decreases (your nose is presumably dragging on the ground at this point) your rudder will lose authority and differential braking would be the way to go.
Differential braking would still be available for directional control, and the rudder would still be effective at the same speeds as normal (effectiveness decreasing as speed decreases). To what extent one would have the presence of mind to use differential braking after the perhaps unexpected collapse of the nose gear is a question. If the engines (assuming a multi-engine aircraft) weren't affected by the gear collapse, differential power would also be an option, although that's not always something that can be fine-tuned, and again would require the presence of mind to use it.
I believe there was a Jet Blue aircraft that landed in LAX a few years ago with a nosegear that, while not collapsed, was useless by virtue of being cocked sideways. My recollection is that they stopped fairly well centered on the runway, although they had the advantage of knowing the situation in advance and working out a plan for exactly what they had. So one might conclude that an operative nosewheel steering isn't essential to staying on the runway, but there's not much doubt that it certainly helps.
