Can Villagers use fence gates?

No, villagers are not able to use fence gates. I tested this in creative, and they always used a long route with a door rather than a short route with a fence gate. I even blocked off the door and they just stood there.

No. Villagers can only open and close wooden doors, but they cannot open and close fence gates. You can, of course, set up a fence gate, but the villagers may try to escape through that opening when you open the fence gate. A better way to trade with villagers is to set up a booth-like thing:

F-Fence V=Villager



FFF  FFF
FVFFFFVF
FVVVVVVF
FFFFFFFF

This way, they cannot escape and you can still trade with them.

No, they can't.

They cannot use any mechanism without the help of redstone or stuff like pressure plates for wooden doors.