Have you had him to the vet yet? They can help with skin issues, suggest other treatments for him. I don't know about back problems. If he's wrestling with your Chi, and being ok with that, maybe he's just trying to figure out what's expected of him? I don't know, he seems like he's trying to be a good boy though, and he's getting along with your other dog.
I'd take him to the vet for the skin issues, for sure!