You can put the amount you pay a sitter or daycare as the work related child care. Your mother does not necessarily need to claim it on her taxes unless it’s a large amount and you file it on your taxes. The courts do not need to know all that information. Usually, they will take your word that this is the amount you pay. If they need proof, maybe you can show them canceled checks?..
The child support is based on the number of overnights with each parent, the incomes of each parent, insurance and child care that is paid out by each parent. If you need to have the children in “day care” more days, then the amount you pay for day care would increase therefore your support amount would decrease. This normally makes the other spouse responsible for a little bit more. Think of it that each of you have to contribute to the total amount the state says it takes to raise that child per month. If the amount you contribute increases, the amount the other spouse decreases as long as the total amount is met.
If your ex gives up her parental rights, which I am not certain can be done without a court order, then she will have no visitation, no rights whatsoever to the child/children. You will have sole, primary physical and legal custody and she would be completely out of the picture. Yes, she would no longer be forced to pay child support, because according to the courts, she would no longer have children…
Giving up parental rights is not the way to decrease child support and I do not believe that the courts would allow it.