Being listed on the birth certificate or not is not the mother's choice. The father can go to the Juvenile and Domestic Relations District Court and have paternity determined and then have Vital Records add him to the birth certificate. Even without being on the birth certificate, the father can petition the court for custody and/or visitation.
You have to be separated in Virginia for one year unless you have a written Agreement. With such an Agreement, the waiting period is reduced to 6 months. The Agreement can and should provide an allocation of debts and assets and support. If you cannot agree, then a court will decide.
It makes no difference which of you actually files for divorce,
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.
The judge will decide the case without mediation, in all probability. Either party has the ability to opt out of mediation.
In Virginia, courts use Child Support Guidelines to calculate child support. The judge must apply these guidelines except in extraordinary cases.