It would be legal to keep the car registered in NC. This affects the insurance on the car but I have heard of other states doing this because NC has one of the lowest insurance rates.
As far as the paper he wants you to sign, it’s legal for him to ask and if you sign you will be giving up your right to 1/2 the marital home & property. He can ask for anything he wants, it does not mean you have to agree.
I suggest that you do not sign anything until you at least speak with an attorney. Never sign anything that makes you uncomfortable and make sure that you understand anything that you do sign. What he’s asking is for you to sign a “separation agreement” giving him all the marital assets except the car which will be yours. This means any bank account funds, retirement funds, equity in the marital home, other property/furiniture…everything is his.