After a DNA test has been established and that father is not on the birth certificate (nor has any custodial rights), can he still has rights?

If father is not on the birth certificate, but he has sufficiently proven that he is the father through a paternity or DNA test, it is his burden to bring an action to the court to amend the birth certificate. Once this is done, he may have rights to visitation, if you have proof he is not fit, you need to present it at this point. However, without a court order you have no obligation to allow visitation, and once he subjects himself to the jurisdiction of the court in an action to amend the birth certificate, he subjects himself to child support. De[pending on the age of the child, and his income, he may not want to have the burden of child support.

In the situation outlined, unless there was a marriage at the time of the conception, the party on the birth certification who is not the biological father would not have either rights or responsibilities for the child but I would want that backed up with a court order.

Once he has been established as the biological father, there should be an order to add him to the birth certificate and (assuming he wants visitation), the court will hear what he believes is best for the child. At that visitation hearing, the mother will also be able to give her opinions, the history, and present evidence and witnesses as to why she believes he should have minimized (or even zero) visitation. Making sure witnesses will be heard and evidence will be admissible can be complicated, you will want to hire an attorney for that, or make sure you have done all your legal research.