His effect sais that if Wolverine would become stunned you could exaust and move him to the back row. Can you do this while attacking?

I know he is already exausted but some effects for exausting thing can be used on already exausted charactors.

Yes. The exhausting is part of the effect, not part of a cost, so you can use it even if he is exhausted.

If you must exhaust as part of the cost, then the character must be ready in order to exhaust it to pay the cost.

If you must exhaust as part of the effect, then it doesn't matter if the character is ready or exhausted beforehand.