Supervised visitation means that the non-custodial parent may not spend time alone with the child. It usually also means that the non-custodial parent may visit the child at a particular time and in a particular place.
No. Visitation rights may not be denied to the non-custodial parent, even though the non-custodial parent is not paying child support.
You may seek a court order defining your visitation rights. Violation of this court order will result in the custodial parent being held in contempt of court.
Generally, the natural parents will have a presumptive right to custody. Only in cases where the parents are found to be unfit, or there are exceptional circumstances, will third parties be granted custody. At any time after a divorce, grandparents may petition the court for visitation rights.