Is Nigma given the power action when he recieves the token in his man reborn ability? Or do you clear it at the end of the turn he was given the token because he wasn't actually given an action?

The adjacent character is the one given the action. No action is given to Edward Nigma to use A Man Reborn? That said, Edward Nigma does not clear at the end of the turn if he was given an action token through A Man Reborn? this turn.

From pp. 6-7 of the Blackest Night Rulebook:
Clearing actually has nothing to do with whether a character was given a non-free action or not. It has to do with whether a character was given a token.

Let say batman is next to edward nigma. It would go like this.

Turn 1. Batman takes action, passes token to nigma. Roll die to see if batman gets hurt as per Nigmas power description
Turn 2. Batman takes action, keeps token. As long as nigma does not take a tokenable (non-free) action he clears
Turn 3. Batman takes action, passes token to Nigma. Roll die to see if batman gets hurt as per Nigmas power description. Because Batman has not taken a token this turn he clears his token.
repeat

Nigma does not actually take an action to receive tokens from an adjacent character. It just happens as part of his power

So, the damage is applied after the action resolves if the one is rolled?

Yes. The action token is applied after the action is resolved so the die roll is done after the action is resolved and any damage would be applied after the action is resolved.

From p. 6 of the Blackest Night Rulebook: