First, I need to let you know I'm not a lawer. But my advice is based on what I have experienced myself.
As far as I know, (According to Georgia law, though...), if the father doesn't legitimise (meaning he has to file a special paperwork through court to have judge decide he is the father) the child who was born out of wedlock, he doesn't get recognized as the father. Another words, he has no rights whatsoever regarding the child's life. So I would say yes, you should have a good chance keeping the full custody. (Espacially when he is a sexual predetor)
As far as SS#, you should get a credit report on her first. You will need to write to those credit agencies like Trans Union, Experian, Equifax. If you see the evidence of an identiy theft (She should have none on her credit since she is a minor), call police. Then they will assist you as to what your next step is. You may also contact federal trade commosion to report the fraud. (www.ftc.gov). These were the steps I had to take when my husband's ex girl friend used my step daughter's SS# (since she was 2 yrs old) to get apartments, phone lines, etc.
Also, if you aware that your ex lives where he is not registered, you need to call police to report it, not only to help yourself in the court in case he tries to get custody, but also to protect other people from being victomised.



