Legal Question in Family Law in Florida

So long as the father doesn't object, the mother can give temporary custody without a problem, though I'd be worried that the father would file a paternity case and ask for custody.

If the mother wants the child to be adopted, the mother should get consent from the father or seek to terminate the father's parental rights.

The birth certificate is irrelevant. If the father denies paternity, the court could order a paternity test, but it's usually not done in adoption cases. In adoption cases, the court usually accepts the mother's word as to who the father is.