Is my ATmega broken, and would replacing it with another break that one, too?
Besides breaking a pin while removing the IC from its socket, or static electricity during manipulation, I don't see what could fry the ATmega. 

However, it is possible that the R2 ATmega has a different bootloader and that may be an issue.

If I was you I would just remove the ATmega from the R3 board and test it on a breadboard (don't need a lot of components to do that); at least that could tell you if it is fried or if it is the R3 board that is fried...