Is there a way for a father to sign over his legal rights before the baby is born? (This is at his own request)

Nothing can be born until the child is born. You then can go to court and try to get your rights terminate but it is rare that this would happen.

The mother of the child can still take you to court for child support and even if your rights are terminate you can be responsible for paying child support at least until the child is 18.

No, before the baby is born there is nothing a father can do. And to be honest it is almost impossible to sign away the rights after the baby is born. And even if you do, unless the mother has a new husband who adopts the father is going to pay child support anyway.

So just don't list him on the birth certificate if you don't want to, then he will not have any rights unless he goes to court for them.

And you can not get child support unless you take him back to court to prove he is the father

There isn't even a way for him to sign over his rights AFTER the baby is born. Parental Rights have to be terminated by a court and courts will generally only grant a TPR to clear the way for an adoption or if the parent represents a danger to the child.

The mother can choose not to name the father on the birth certificate which would mean he has no rights or responsibilities unless challenged. But this would also prevent filing for child support.

If you should apply for public assistance, then they will require that you identify the father and they will go after him for support.