Possession
In regards to your question of custody, once your baby is born, you will not have custody. You will have what is known as "possession". Custody is awarded by the court. I know this because I was sued for custody for my daughter by her father. We had never been married, and his name was not on the birth certificate. I was informed by an attorney that if he wanted to leave the state with my child, he could because he would then be the parent of possession. In order to ensure that I had the legal right to my child, I had to file for custody, which I did and won.