What rights does a dad have if he is not on the birth certificate and can't prove he is the dad and he also doesn't pay any child support

He can file asking for visitation or custody and DNA testing would prove if he is the father or no matter what you claim or what's on the birth certificate.

Of course at that time he would in all likely hood start being obligated to pay support if he doesn't get full custody.

You have a choice... keep it as it is where nothing legally indicates he is the father and of course he won't have to pay... or you can establish him as the father legally and he WILL have visitation rights and even have the right to go after custody... but he then might be on the hook for support... or even possible put YOU on the hook for support if he gets custody, particularly if YOU have the better paying job.

What makes you think he can't prove it? If he applies to the court for custody and visitation, the court will order a Paternity test. Such test (usually DNA nowadays) are conclusive proof. So once he is confirmed the bio father, he can get custody.

If you go to court for child support, he can ask for a paternity test to confirm before a court will order support.