If you and the other parent live together, either one of you (but not both) may claim the child. You may decide between you which one will claim the child. Only if you can’t agree, do the IRS tie breaker rules apply (higher AGI), to see who has first choice. It may be worthwhile to prepare trial returns, both ways, to see which way the family comes out best. This tool may be useful: https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1.
To use Head of Household (HOH) filing status, you must be the one claiming the dependent and the one paying most of the household expenses (you can't split the expenses exactly 50-50).
For calculating the dependent care credit (DCC), you may not count the money she reimburses you for baby sitting. Only the parent claiming the dependent can claim the DCC. Baby sitting expenses are not part of household expenses when determining eligibility for HOH filing status.
