I didn't read your whole post, it was a bit rambling, but I don't know why it would kill you to let the kids be with him on new years day. You get them almost every day of the year otherwise.
From the tone of the post it sounds like you are pissed at him for moving away and not paying the medical expenses and you are punishing him by making your stand about when he can and can't have the kids. He will probobly not file court papers (unless new girlfriend has money), but will back down and back even further out of your kids life. You are hurting your kids more than you are hurting him.
Sorry if I took your post wrong, but from what I read and could understand, these are my thoughts.