That's not the way to interpret it.

The power action given to the non-Doppy figure is simply an action. You cannot give the non-Doppy a power action and have that activate HSS and Doppy's SP.

beast boys card reads give beast boy a move action that deals no pushing damage. if he gets a second token and he doesnt have will power or starts off next to a higher character with same team symbol and inspiring command. the he will take pushing damage and wont be able to use the morph ability. if they wanted him to not be effected by pushing then they would have given him indomitable or said on his trait that he had willpower. beast boys power is writen this way i believe to show that he cant turn into another form when he is exhausted

No, he won't take pushing damage when he uses Morph. That is the point of his Morph trait.

The problem with doing that is that if they'd done that then he would not take pushing damage for attacking or doing any other non-free action. As it is written, only move actions result in no pushing.

That may be what you believe, but I can assure you that is not the correct ruling.