You are only able to claim Head of Household and the child care credit in the years that you aren't claiming the child if 1) you're the custodial parent and 2) you paid the cost of the keeping up the home for more than half the year.
The custodial parent is the one that the child spends the most nights with. Even though you and your ex have 50/50 custody, one of you likely would have spent more nights with the child. If it was truly even, then you wouldn't meet the "paying the cost of keep up the home for more than half the year" test.
If, based on the information above, you determine that you qualify for head of household and the child care credit, you'll need to mail your return to be processed. The IRS will then request documentation for you to prove that you qualify.
