Even if the father's name is on the birth certificate, and is living with the mother, except in Arizona.
No she cant be the mother or have custody for a child she dont even own
No, you put the father on the birth certificate! Then it wouldn't be a birth certificate!
No, as he has no legal rights to the child even if he sign the certificate himself. You will need to file for custody. see links below.
Not if he has visitation rights or shared custody.
It is possible to legally establish paternity even if the father's name is not on the birth certificate.
The father must sign to add his name to the birth certificate.
Except in Arizona, the mother has sole custody and the father has to apply for his rights through the courts.
Even if no custody and access orders are in place, the father has up to six months to file an injunction ordering the return of the child to the jurisdiction of the court.
When he signed the birth certificate he became the father until/unless the courts rule otherwise.
You can list anyone you want on the birth certificate. Its up to the court an a paternity test to decide who really is the father.