I highly disagree with your theory for multiple reasons op.
I think he has it because he either opened his third eye, or to represent neutrality towards all 5 nations. He did a lot of bad so nobody will except him, but because he helped save the world, they'll allow him to live in peace on his own.
Naruto's hand is bandage because of hitting suckske in the head so much to beat some sense into him suckske's head is bandaged from taking the beating.