For child 1:
Only you have the automatic right to claim the child as a dependent. Father 1 does not, because the child does not live with him, and father 2 does not, because father 2 is not the biological or legal father of child 1. If you won't be filing a tax return, or you want to allow father 1 to claim the $2000 child tax credit, you can give father 1 a signed form 8332 which releases child 1 to father 1. If you do file a tax return, you can list child 1 and answer yes the child lived with me more than half the year, yes there is a custody agreement, yes I will allow the other parent to claim the child. That will create a status in Turbotax where child 1 will not claim the $2000 credit for you, but will qualify you for EIC, the dependent care credit, and head of household, because those things can't be waived or transferred even with an agreement or court order. (Or, if you decide to claim child 1 yourself, you will answer yes you had custody, but no, there is no agreement to release the child.)
For child 2:
We need to know if child 2 and father 2 lived in the same home more than half the year, or more than half the time since the child was born if the child was born in 2022.
If father 2 did not live in the same home as child 2 for more than half the year, only you can claim child 2 as a dependent. You can't release the tax credit with form 8332 because that only applies to parents who live apart, not parents who live together.
If father 2 did live in the same home as child 2 for more than half the year (or more than half the time since the child was born) then either you or father 2 can claim child 2. If you can't agree on who should claim the child, the first tiebreaker is which parent did the child live with longer and the second tiebreaker is which parent had the higher income. But if you agree, then whoever you agree can claim child 2. The parent who is not claiming child 2 should not even list them on their tax return, to prevent confusion because the interview questions can sometimes be misunderstood. The parent who does claim the child should answer yes the child lived with me more than half the year, but NO, there is no custody agreement. (This is the correct answer in this situation because the question about a custody agreement only applies to parents who live apart.)
Whichever person (mother or father 2) paid more than half the total household expenses for the year can also file as head of household if they claim at least one child as a dependent. The parent who paid less than half the household expenses must file as single regardless of how many children they do or don't claim.
