Can the father take the child if he didn’t pay support?

My daughter is 19 years old with and 11 month old baby. The baby’s father (23 years old) walked out on her after the baby was 3 days old. She has not had any support whatsoever from him, no child support, nothing. He gave her 1-2 boxes of diapers and that’s about it. The two of them were living in the same state at the time of the baby’s birth. My daughter moved 9 months ago to another the state to be closer to her family. The guy moved back to his mother’s house. Before moving, he filed a paternity suit (after my daughter had moved) and it was presented to her father on her behalf. The baby’s father has lied to the courts saying he lived/has lived in the same state the entire time. My daughter hired a very crappy lawyer (she didn’t know at the time), who did not support/defend her in court (his lawyer clearly was buddies with the judge) and overruled my daughter’s proof she had of his residency. My daughter wanted the jurisdiction changed (as she had been living there over 6 months at the time, but they refused). She has been ordered 3 times now to return on her own dime (and she is not employed and has no money, has been staying at a relative’s home) all of which has been an extreme burden on her and the baby. The courts are ordering for joint custody with him (who doesn’t even seem interested in being a dad, its Pedro’s mother who is pushing for all of this). My daughter never agreed to or signed anything stating that she would agree to this joint custody arrangement. He has all along stated he wants the baby’s name to be changed to his last name, and he wants the baby’s first name to also be changed - and this has been a demand of his from day one, which my daughter has refused rightfully so. Additionally, he has threatened my daughter in the past, stating, ‘I could take the baby to Columbia and you will never see her again.’