A: If you are not married to someone else, the father can sign the birth certificate and be listed. You need the father's consent/signature to name him as the father, unless you want to initiate legal action. If he doesn't admit paternity and sign the birth certificate- you can establish him as the legal father via a child support action or paternity action. You could seek to amend the birth certificate to name him as the father and to change the child's name.
A: You can't legally do anything about where he chooses to live. However, if you have children together and/or you are entitled to alimony, you can seek support as of the date he left, without filing for divorce.