The child can do anything he wants to :)
How does he have any visitation rights with a custody and child support order?
Only the court can legally suspend visitation rights.
If there are no court orders both parents have equal rights.
no
An unmarried father cannot "choose" to not pay child support. The laws in every state require that a father pay for the support of his child. The mother must bring an action to the appropriate court so that a child support order can be established. Visitation rights are separate and a father can have visitation rights established by the court. Visitation rights are not dependent on paying child support.
If he doesn't take away visitation rights he should
The father might seek visitation and/or custody.
Generally, no. However, you have not provided any detail and there may be circumstances under which an unfit mother may lose custody to a step father who has been a stable parent to the child. You should speak with an attorney who can review your situation and explain your rights and options.
No. Not if a family court has jurisdiction over the child and the father has visitation rights. If the parents were never married and there has been no court involvement or paternity established the mother can try to move. However, the father could file an action in court to establish his paternity, visitation rights and prevent the removal of the child from the state.
A judge can restrict visitation rights if the judge feels that it is in the best interests of the child to do so.
Your child does have the right not to be punished by you for something the father is not doing.
He has the right to petition the courts for visitation.