Fair Play: Each time an action resolves in which an opposing character has used Outwit targeting Mr. Terrific or an adjacent friendly character, Mr. Terrific may use Outwit as a free action, but can't counter any game effect that allows a character to use Outwit.

If Mr. Terrific outwits something with his trait, when does the countered power return to the character?
As the power Outwit says: "until he beginning of your next turn." So, on your opponents turn, they use Outwit on a qualifying character. Mr. Terrific then outwits something. The opponent's turn ends and Mr. Terrific's turn begins and the countered power returns.

Can Mr. Terrific use this trait even on the clicks where he doesn't have Outwit?
Yes. This use of Outwit is independent of the standard power. In fact, because the power says "Each time...", Mr. Terrific is able to use the Outwit from Fair Play more than once per turn.