Assume I'm a level 1 Thief. Is there a limit to the number of times I attempt to steal something? Can I just attempt over and over again until I get the thing? Does it even have to be on my turn?
There's no limit on attempts per turn, but you do have to discard a card to attempt it, so you can't just attempt forever. From the FAQ:
Q. How often can a Thief steal?
A. As long as he has cards to discard.
Q. What happens to a Level 1 Thief who fails an attempt to steal? Does he die?
A. Nothing happens to him. You can't go below Level 1. (Beware the Level 1 Thief – he literally has nothing to lose!)