Accuracy is needed for turn meter reduction from skills. Don't know about Evil Eye.

I think Evil Eye doesn't need accuracy, but it only can happen once per enemy per battle per champion

I too would like to know for sure. Since turn meter manipulation is not technically a buff or debuff, accuracy and resistance should not be factors. I cannot find anything official about this though.

Masteries do not need accuracy. Everything else non-damage-related that affects the other team does, with the sole exception of Block Revive.