It is not possible to say with any kind of certainty what the total cost will be. That is why lawyers do the Retainer Fee (deposit against final cost) and hourly rate method. It depends on the time spent. Much of that time may be spent in the useless, but common, wait at the courthouse for your case even to be called to be heard. The ultimate time and cost often depend on the other party, the other party's lawyer and the judge. You have no control over these factors.
The decision whether to require supervised visitation is up to the judge and depends on whether the judge thinks that the child is at risk.
The biological father does not "lose" rights. They can be terminated in an adoption, but not lost by not paying support. You can file for support through the court or the Division of Child Support Enforcement.
Courts hope that the parties can agree on a visitation schedule without the need for a judge to put one in place. If you cannot agree, you can go to court and ask for specified visitation. Until then, the custodial parent decides what is reasonable.
You can file to have the Agreement enforced as a contract and sue her for breach of contract or file for divorce or file to have the Agreement made into a court order and then enforced using the contempt power of the court.
I strongly suspect that any attempt to secretly DNA test will not lead to admissible results; however, those results could be used to have the court order an "official" test. Why wasn't this brought up in the divorce case?