Paying child support will not cause the father to lose his parental rights - neither will not paying 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.
if the mother terminates her rights can he collect child support from the mother if child lives with him?
Termination of parental rights does not, in itself, terminate child support.
How does he have any visitation rights with a custody and child support order?
Signing rights away will not stop any child support from being taken from the father. If the child is the fathers then he is financially responsible for the child weather he has rights or not.
As you have now given up your rights to the child you do not have to pay back child support.
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.
No, when a mother or father relinquishes his parental rights he is no longer responsible for child support.