Tennessee requires that minor name changes have the consent of BOTH parents and be in the best interest of the child. In fact, you will be served with notice of the hearing, and your assertion that you do not want the change will derail it.

It will not have anything to do with your parental rights even if you consent to it.

Custody is a completely different issue. Support is a completely different issue. None of the three issues are cross-linked to another.
If you had custody, the court isn't going to rescind the support payment order. For the period of time that you did not have custody, they may vacate the payments.

The name change is simple. Just go and oppose it.

If she starts custody/support suits, you would be advised to get a lawyer.