Critical Chance only for basic attacks?

Basic attacks and some certain Skills, like Gangplanks Q. If you want to know if your favorite champ can crit with skills look it up in the wiki, they have practically everything there :D

I think all of the abilities that can crit specifically say that in the skill description in-game. Normally, it's only basic attacks.

Anything that applys on hit effects or is a "Basic attack modifier" such as Nasus Q. Not sure if this applies to Hydra