Very briefly:
If you live apart and share custody, only the parent where the children live more than half the nights of the year (183 or more nights) is automatically entitled to claim the children as dependents. The IRS does not follow court custody orders, only where the child actually physically lived. In some cases, the custodial parent may allow the non-custodial parent to claim one or more children as dependents. This requires that the custodial parent give the non-custodial parent a signed form 8332 dependent release, that the non-custodial parent must physically mail to the IRS after e-filing.
In this case, only the custodial parent can use the children to qualify for head of household and the child and dependent care credit. However the child tax credit and the additional child tax credit go to whichever parent claims the child as a dependent.
