If you were the custodial parent who the children lived with more than half the year, you are allowed to claim the children as qualifying children for certain tax benefits. Those include head of household filing status, earned income credit and child care credit. If you choose to or if required by divorce decree, you can relinquish the dependency exemption and child tax credit to the other parent by providing form 8332 or similar statement. The tax benefits cannot be split any other way.
If you are the custodial parent but are releasing the dependency exemption to the other parent, the description for your child in Personal Info would say Nondependent for EIC and child care credit only. If you are not releasing the dependency exemption to the other parent, it would say Dependent.
Unfortunately only the custodial parent can claim child care expenses so child 2's father will not be able to claim the child care credit on his tax return even though he paid part of the expenses. You can't let him claim child care expenses if you are the custodial parent. You might consider giving him part of the credit you get.
When you enter a child who was born during the year on your tax return, you indicate the child lived with you the whole year. That is the correct answer for a child that lived with you all the time since the day the child was born.
