You can schedule a hearing to have the contract (which is what the Agreement is) be made into a court Order and then have it enforced by contempt. You also no longer need to have court hearings for uncontested divorces. Virginia authorizes the use of Affidavits effective July 1, 2012. Your money would be well spent to consult with a lawyer.
If the case is being heard in New York, then they are applying New York law. Under Virginia law, the court can award support from the date of filing forward. It cannot go back before the date of filing for support. The relationship between the parent and child, however, has no bearing on the duty to provide support.
You can petition the court to change the child's name. The father would be served with Notice, and he could appear in court and contest the name change or he may choose not to appear. If he does not object, a name change order will be entered.
From a legal perspective, the advantages of marriage often can be achieved through other means-but these means also require a "piece of paper'. Through marriage, you acquire inheritance rights. Most of that can be achieved with a Will, but Wills can be changed and even if a spouse tries to disinherit the other one, they still are entitled to one-third of the deceased's estate if they are not separated. A spouse is the one who makes the medical decisions when you cannot. This can be achieved...
Does your child already have a passport? If so, you do not need the other parent's consent. You would give the required notice of the trip. If he objected, the burden would be on him to ask the court to prevent the trip.
Have you and the father considered going to mediation? This is the kind of issue that would be resolved best without the need for litigation, if possible.
If custody is at issue in a divorce case then the mental and physical condition of the parties are statutory factors that must be considered by the court. For a very brief period, there was a statute in Virginia that made mental health records exempt from disclosure in such situations. That is no longer the case, and the situation is as it always has been. The records can be subpoenaed and a court will compel their production if there is a custody issue.