If your child's father pays child support but has not had any contact with her in a year does he still have any rights to her?
Yes. He has rights until his rights are terminated by a court order.
What rights does a father have if he abandons his child and unwed mother does not pay any support or have any contact with child?
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.
No, you can't. First, you can't sue a parent for child support. Second, there is nobody to collect it from since your father has passed. However, if your father has an estate you could contact the attorney handling the estate or a private attorney to determine if you have any rights in his estate as an heir at law.