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DearCustomer- In order to change any custody or visitation order you must file a motion with the court that issued the current order. Child support is a separate issue. If he is not paying you can file a motion for contempt. Unless the father is abusing the child or is some danger to the child it will be difficult to remove him from the picture if he wants to be a part of the child's life.
I understand your frustration but as long as he's the father he has parental rights and unless he will voluntarily relinquish them you can't force him out of you and your child's life.You can ask the court to limit the visitation etc but to completely remove him is much more difficult.
Well then he has no rights until he files for paternity and establishes visitation. I assume you were never married and there was no divorce decree so if there's nothing from the courts with respect to the child you can simply deny him any contact. Of course he can file for paternity etc but then there will be a court order for support issued so he may think twice about that idea.
Well it appears there is a court order for support so therefore he doesn't have to prove paternity. All he has to do is file a motion for visitation however you can contest that at least to the extent that you would request only supervised visitation. You should also file a motion for contempt of non payment of support. While child support and visitation are separate issues you deserve the money and you should hold him responsible.
That's probably the best plan. You have no duty to offer visitation until there is a court order. Assuming he is not trained in the law he will probably need to hire an attorney to accomplish what he wants.