Courts do not generally allow a parent to terminate parental rights unless there is another adult waiting to adopt the child. You need to add more detail to your question since you have suggested an unusual arrangement where you terminate your rights yet still pay child support. You should consult with an attorney. If you plan to continue to pay child support one wonders why you want to terminate your parental rights.
He will still have to pay support until/unless the child is adopted.
Termination of parental rights does not terminate one's child support obligation.
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.
No he does not because he is not the child's parent anymore.
It's likely that the father will still owe support until/unless the child is adopted. see links
yes you can but you would still have to pay for child support
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.
Yes, a father can give up his parental rights but he still have to pay child support.
As the father has no assumed rights to the child, he can still be ordered to pay by the courts.
yes
Yes. He has rights until his rights are terminated by a court order.
no the mother should have the legal obligation to take care of the child