To my knowledge there is no standing child support order and the original was thrown out because they lived together... she(custodial parent) is continuously threatening to take out an abandonment warrant but his name is not on the birth certificate the child has not been legitimized and he pays her $45 a week for daycare... what are his options? Can she file the warrant or not?

In GA they refer to failure to support a child as abandonment. Really it refers to non payment of child support. But before one can apply for support, paternity needs to be established. So she can't get a warrant until she establishes paternity which involves a court hearing.

However, once she does get paternity established, she can then apply for child support.