My opponent's AvX Emma is on click #5 and has normal Outwit. She has Outwitted my AvX Spidey's Outwit. It's my turn, and another character on my force just dealt 1 damage to Emma, putting her on click #6, which does not give her normal Outwit, but instead gives her a special power that grants Outwit with a range of 7. Does Spidey get his Outwit back?

I would imagine spidey gets it back since its not the same outwit

I would say no since the power does not specify if the outwit is lost. Unlike perplex which says what happens

Spidey gets it back for the reason SuperSaiyan5Swagger stated.

Another quick Outwit question. If an opposing character outwits a power that is showing on the dial of another character, but the same character being outwitted was granted the same power from a resource/relic; is the power still outwitted? (IE..Jane outwitts Joe's charge that is showing on the dial, but Joe is also equipped with a hammer which states that Joe "can use charge")

The charge granted by the hammer would kick in. Outwit only removes a specific source for the power/ability.