Ideally, your divorce agreement would specify, but yes, if it is your year you take her as a dependent, claim head of household, and don't worry about the 50/50 custody.

You can't claim his daycare payments.

My ex and I divorced last year, he claimed our DC last year, this year it's mine, per our agreement to claim DC. My questions are
1) Do I state DC lived with me the whole year despite sharing her 50/50 since we alternate claiming? Otherwise it seems I cannot claim her as a dependent.
2) Can I claim the money we both paid for her daycare on there rather than just my half? So we can get the max deduction since only once of us can claim her? How does that work? Because if he cannot claim her, there's several thousands unaccounted for, and that seems like a waste. Is it wrong/illegal to do so?

Thanks