Always use rudder to keep the slip skid ball centered. Except when taking off and landing in crosswinds, that's all you ever do with it. If the airplane rolls with the slip/skid ball centered, that's a bummer because it needs some aileron trim and there is only a fixed tab that you can't adjust until you're back on the ground.
For adjustable trim, the Archer has a trim tab on the stabilator surface for pitch/speed trim (doing double duty as an anti-servo tab to provide desirable pitch feel forces), and a bungee (spring) trim for the rudder control, operated by the black knob below the throttle quadrant by the passenger's left knee. Here, we are only really concerned with the rudder trim.
The trim bungee provides a mechanical centering force to the rudder cable circuit that can be adjusted within a small range each side of actual center. When you're flying, you crank the knob to center the skid ball.
If the skid ball is to the right, requiring you to hold right rudder to center the ball, you crank the knob clockwise, which squeezes a bit of right rudder for you so you can stop squeezing the pedal with your foot (you could take a bungee cord from the hardware store, hook it over the left rudder pedal, and pull some tension on it to apply some right rudder input - it's pretty much what the internal trim bungee is doing).
So in general, you just twiddle the knob as required to center the ball so you can relax your feet. If the ball is off center, you can expect the airplane to want to roll away from the ball (ball to the right, left rolling tendency).
The Piper Cherokee family has a fairly strong rolling tendency when skidding, which is why they put the rudder trimmer in there in the first place; most light planes don't bother with rudder trim except for a ground adjustable only tab on the rudder that is normally bent (by trial and error) so as to center the ball in cruise flight.
When you have the ball centered nicely with no foot pressure with the trim knob, and you make speed/attitude changes, and power changes, it tends to change the equilibrium point in yaw so you will find that significant changes make the ball move from where you've trimmed it.
If the change is temporary and you'll be returning to your original power setting and speed, say in the next minute or two, you don't bother and just squeeze rudder with your foot knowing that you can relax it in a minute or two, but if the change is longer term, just turn the knob again to center the ball so you don't have to hold pressure with your foot.
So to answer your questions:
- You should be always holding whatever rudder is needed to center the ball. Without the Archer's rudder trim, you would have to hold this rudder pressure continuously except at one particular power and speed. So 3 cheers for the adjustable rudder trim. Use the trim for any changes that are more than temporary so you don't have to squeeze the pedal all the time.
- In the Cherokee family, keeping the ball centered will minimize rolling tendency, so it should fly straight hands-off if you have the trim set to center the ball. If the airplane rolls with the ball centered and needs aileron pressure to fly straight, in the normal cruising power setting and speed, the fixed tab on the aileron needs to be adjusted (bent up or down as required).
At some point your instructor will start having you practice using the rudder trim knob. They're just waiting until you've internalized some of the basics first.