Does the father to my child have any rights to his son since he is not on the birth certificate?

Yes, the bio father always has rights, he may have to go to court and get a DNA test to prove he is the father

Being on the birth certificate just means he does not have to prove it.

First its not a good idea to piggyback your question on someone else's. This can lead to confusion. You should start a new thread. So I've moved your question to its own thread.

The father has the same rights as any father, but without being on the birth certificate, he has to go to court to get them enforced. He has to ask the court to declare him the legal father. A paternity test will be ordered and it it comes back positive, he will be declared the legal father and have the same rights as any father.