No, but he shouldn't try. He can file with the court to give up his rights, but it requires the approval of the mother, and she cannot be on Welfare, now or in the future. If she is, she gives up any right to child support and to making this decision.
Paying child support will not cause the father to lose his parental rights - neither will not paying child support.
only if the child is then adopted
That is the case in most states, once you sign over the rights to a child, you are no longer considered the guardian and have no legal or financial obligations to that child.
The right to petition the courts for the right to see and support his child.
How does he have any visitation rights with a custody and child support order?
Termination of parental rights does not, in itself, terminate child support.
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.
Generally, no.
no
You can sign your rights away but you will still have to pay child support if you are the father or mother of the child. There is no way to avoid paying child support.
Termination of parental rights does not terminate one's child support obligation.
He is still the father with/without child support and has rights. Take him to court to get the child support and make arrangements for your move to a different state.