They are not stepchildren if you were never legally married.
If they are not related by blood or marriage, then they are unrelated people. You can only claim unrelated children as dependents if:
1. no other "taxpayer" can claim them as dependents,
2. they lived in your home as a member of your household for the entire year, and
3. you paid more than half their support, and
4. their taxable income is less than $4700.
This type of dependent will qualify you for a $500 credit for "other dependent", not the $2000 child tax credit.
If the children lived in the other parent's home for some number of nights, then they didn't live in your home the entire year.
However, if the children did not live with the other parent more than half the nights of the year, then the other parent can't claim them either, at least, not as "qualifying child" dependents. The other parent could claim them as "qualifying relative" or "other" dependents if the other parent paid more than half their financial support, or if the other parent gets a signed agreement from everyone who paid more than 10% of the child's support (that would be you) agreeing that you will not claim them and the other parent can. This is called a "multiple support" agreement and is found in publication 501. This type of dependent would not allow the other parent to file as head of household or claim EITC.
https://www.irs.gov/forms-pubs/about-publication-501
Also, it is important to mention that the IRS does not recognize court orders, and there is no such thing as 50/50 custody in a year with an odd number of days. The IRS only recognizes where the child physically spent the night. You can't even claim the biological child you share together unless that child lived in your home more than half the year, which means the child slept in your home 183 or more nights. (For sleepovers with friends, vacations, and visits to grandparents, you get credit if it would have been your night without the activity.) If you did not have physical custody of the child in your home for 183 or more nights, you can't claim them even if the court says you have 50/50 custody. You may need to get a calendar and count the actual nights.
