Can a father in PA sign over his rights and give sole custody of the child to the mother?

Custody and Support are separate issues. A parent can reduce their role in custody but that does not relieve the parent of the obligation to provide support. The court will deem a level of income to your ex and may even find him in contempt if he continues to fail to pay support. If he is getting unemployment or disability benefits then the court will attach such benefits to help pay the obligation. For a parent to relinquish parental rights, another would be expected to step in to fill such responsibility, such as a step parent wanting to adopt the child.