The three axes of motion are controlled in this way:-

Pitch - Elevator
Roll - Ailerons
Yaw - Rudder.

In other words, the rudder isn't there to create roll (that's the aileron's job) but to initiate turns to the left or right.

I say 'initiate' because a turn to, say, the left makes the right wing go faster, and therefore creates more lift on that side, so the rudder also creates a banking effect in the direction of yaw. If this persists, (especially if altitude is maintained with a bit of up elevator) then the aircraft will eventually make a U turn as you described.

It's actually more complicated than that, but that'll do for the purposes of this explanation!

If you want to roll, (that is, roll around the axis that runs from the nose to the tail )ailerons will do the job.

A spin (often confused with a roll) is entirely different. It's the result of losing lift from the wing, and letting the control surfaces on the tail take over.

In a full size aircraft, spins can be dangerous, in that there can be a maximum number the aircraft is capable of before it naturally flattens out. You've heard of the expression 'in a flat spin' meaning the person has lost control? Some aircraft, once they are in a flat spin cannot get out of it, as all the control surfaces are in a stalled condition and the pilot simply has no control.
I have seen a Tiger Moth flat spin all the way to the ground, but, as the vertical speed was so slow, the pilot happily survived, as did most of the structure of the aircraft! It came down like a sycamore leaf!!

A 'snap' is really the initial phase of a spin, in that it's caused by loss of lift as the wing starts to stall.

Hope that helped!

Happy flying!

-David C.