I am not real sure I completely understand what you just said, but...
1. If there is no court order for child support and he is just doing it, then he has no obligation to send it. But I would continue to pay something monthly. Here is an online child support calculator for TN which you can use as an estimate of what is fair to send her.
Tennessee Department of Human Services If there is a court order for child support then he would need to go ask for a modification.
2. If he has been determined to be the legal father by one of these methods:
a) Affidavit of paternity
b) DNA
c) name on birth certificate
Then he does have legal rights to her, and to have say in her life, and do things such as be listed as her father on school records etc. The thing is without a court order it is difficult. Most of the time as long as you have something proving paternity you can obtain all the school and medical records.
3. Fathers have great opportunities these days for joint physical custody. The courts really favor it. If you have a court order, file an Order to Show Cause for modification of custody/visitation. If you don't have one, then file an Order To Show Cause for Custody/Visitation to get one. All of this is best done by a lawyer I do not recommend trying it on your own as it can be very messy and complicated and there is a lot at stake.
4. Money will really only play a big role in determining child support not custody. Child support is determined by the amount of time each parent has with the child and the parents incomes. It is normal for the NCP to pay the CP child support for the months of summer/vacations times even when the NCP has the child with them.
Hope this helps.