1. Can he claim you. No, not this year, because your income is more than $5050.
2. Can he claim your children. Maybe. He can claim them if you are not a taxpayer, and they lived in his home the entire year (you didn't mention this one way or the other), and he provided more than half their financial support. They would be "other" dependents and would qualify for a $500 tax credit, and not the $2000 child tax credit. (You are not a "taxpayer" if you don't file a tax return, or only file to get a refund of your job withholding and don't claim any dependents, credits or other benefits.)
3. No. Even though he cares for them and provides support for the home, they are not a qualifying type of dependent for head of household.
You may want to explore filing a return and claiming the children yourself. You would get a little bit of the additional child tax credit, and a little bit of Earned Income Credit, from $7000 of wages. That might be more than the $500 per child he would get. However, you can't split the children. If you claim one to get some kind of credit, you are a "taxpayer" which means you must claim both and he can't claim either, even if you wanted to allow it. You should at least compare scenarios.
