cabudinen :
Not at all, your GPU wont display a resolution that your TV/Monitor does not support.
1) Well, that's not true. It can happen, though it's pretty rare now. It can happen easily if you have a VGA connection though.
However, that would just cause the monitor to freak out, and it would be pretty obvious I'd think. Usually I just got "monitor out of tolerance" or something and it snapped back to the proper resolution if I didn't apply the proper settings.
2) Changing the resolution CAN also cause a game to crash if it's stressing your graphics card more and it's unstable (perhaps insufficient cooling, or overclocked).
*If you have a SPECIFIC question related to a game crashing then give your PC, the game and details of the crash.