"or will it only turn to the right if it is in contact with the surface?" - correct (for 737 anyway)
"Also what happens if you have full right rudder in place" - you generally have more problems than nosewheel steering
I really do not know the answer to that - on the 737 the steering only 'engages' with 'weight on' and whether it zeroes on engagement I know not.
Edit for typo