EIC and Dependent Care Credit

I have 2 taxpayers who are unmarried but live under the same roof with their 2 children. The parents of the children both work outside the home. The father of the children claimed the children in 2023 and filed HOH and the mother claimed S and they share in the household expenses. The father left his employment to start a business in 2024 and did not earn enough income to pay for the dependent care of the 2 children. The mother of the children paid for more than 50% of the day care costs in 2024. 

My questions are:

Can the mother claim the dependent care costs on her filing without claiming the children as dependents since she in essence paid for these costs? The software is allowing the Child Dependent Care Credit of $600 per child. 

If the father does not claim the children this year and allows the mother to do so, the ProConnect software is still calculating EIC for the father's return since his income qualifies for EIC and he does have 2 qualifying children in which he provides a household with the other parent. Is this correct as I know these are his qualifying children but I thought you had to claim them as dependents in order to qualify for EIC? I have suppressed the children as dependents so he does not receive CTC and the mother does but EIC is still calculating and amount. The mother's income does not qualify for EIC this year.  

I reviewed the IRS rules and cannot confirm if the software is calculating correctly.

I would love your feedback. Thanks!   Only a few more extensions to go to complete the season.......