I've been trying to find an answer to the question about damaged being given to a character who does a dble power action according to their abilities.

BUT....if this character has an ability such as indomitable,willpower, or Power Cosmic.....they don't take the damage?

Pushing damage is dealt normally.
Any ability that prevents pushing damage functions normally.

It's just giving the character two tokens.
Pushing damage is normal here.

If the character can not avoid push damage then they take it; If they can avoid push damage they do not.