Do we have to do any paperwork to allow my daughter to be paid to care for her father?

If her dad is paying her out of his own money, it would be wise to do a Caregiver Contract and to withhold taxes, Medicare and FICA. It also shows that the money is a caregiving expense and not a gift, in case Medicaid is needed down the road.

Medicaid has a 5 year look back on spending is most states.