Can the father sign over his rights, and what does he do to get the process started, what forms does he need to do so.

He can voluntarily sign over his rights, he needs to go to the courthouse and bring the mother with him, and they will help you there. He will still be liable for past child support and even if rights are terminated, if there is a child support order in place, it still stands.

If the mother agrees, the judge may allow it. But not always. And of course even if "rights" are given up, unless the custodial parent has a new partner to adopt, the father will still have to pay child support, giving up rights has nothing at all to do with child support obligations.