The children did live with you for more than half the year. They don't have to have lived with you in your own home. As long as you and they lived in the same household, they were living with you.
They are probably your Qualifying Children, not Qualifying Relatives. If one of their parents is your brother or sister, the children meet the relationship test. For you to claim them as qualifying children it doesn't matter who supported them, as long as they did not provide more than half of their own support themselves.
If the father was not entitled to claim the children, then yes, he should file an amended return to remove them. Even if he does that, you will not be able to e-file your tax return claiming them. You will have to file by mail. If he can be convinced to file the amended return that would make the situation much smoother. If he doesn't do that, the IRS will challenge both your tax return and his, and you would have to convince the IRS that you are the one who is entitled to claim them.
The IRS doesn't look at who had legal custody. They look at who the children lived with.
