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Dear JACUSTOMER - I'm not certain why you don't want child support since he would be required by law to pay but he is correct that he can assert his right to visit and any other parental rights. He would first have to establish legal paternity through the courts. Then he would likely have to go through a period of supervised visitation in order to gain full visitation rights. The court isn't just going to turn over a 3 year old to someone she doesn't know.
You can't force him to sign over his paternal rights but maybe if you called his bluff and asked for support he might agree to do so. It sounds to me like he is getting away with intimidation of you by making these threats and not paying any price. If he had to pay support or go to jail for contempt he may be less likely to be doing all these things. If he really wanted to have a relationship with his daughter he would have done so a long time ago.
For now you really can't do anything and if you don't want to file for support then I would ignore him. Unless he goes through the court process he has no rights to your child.
Well the court will want to know who the father is and I would not recommend committing perjury by making a false statement under oath. In any adoption the father must consent. I would think that if he hasn't shown an interest to this point and if he were faced with 15 years or so of child support he might readily consent to giving up his rights.