If the father of my child is not on the birth certificate, we are not married and he does not pay child support does he have any legal rights over my child for the state of Ohio where I live or west Virginia where my child was born

Does he have any "rights" yes and no.

He has full rights but he will have to go to court, ask for a DNA test and prove he is the father, then he can get visits if he wants to.

So he has the rights, but will have to go to court and be proven the father to enforce the rights.