No child support is usally handled by a diffrent judge as to custody is handed by a family court judge but you should be helping pay I know a lot fathers don't feel they should pay the mom or trust the money is going to their child but you don't want to loose your license or go to jail. Your child needs their father and either way it is your child to which also means you have just as much right to see your son just make sure you have joint custody and visitation and if you already do and she's keeping him from you then you can file a violation of Order against her.
No. Visitation is a separate issue from child support. You have every right, if allowed in the court order, to see the child. If your rights to visit the child are being denied you need to file contempt charges with the court. Not paying child support will not stop your right to visitation.
If she wants to be stubborn like that, most females will do that. If you do have joint custody, you can go back down to the court and talk to the judge.