If I mind control A character and move it somewhere and then turn off a power (in this case impervious) upon resolution of the mind control, does the turned off power return upon the return of control of the character OR foes the power remain off?

You have to turn off the power BEFORE you give the fig an action because as soon as the action is resolved it returns to the control of your opponent. But the power would remain off until the end of that turn.
So you can MC a fig with toughness, turn it off, move the fig next to one of your guys and then attack it with that guy and toughness would still be off until the end of your turn.

Except you may only turn off Optional powers. Toughness and Invulnerability are not optional, the only damage reducer that is optional is Impervious (until July, at least... then none of them are).

Check the PAC. If you see this, (optional), after a power name, the you can turn it off. Otherwise, powers can't be turned off at all.