I am sorry to hear that. Hopefully you guys can sort that out.
The legal situation is this: If the baby is born in Germany German law will rule. Under German law custody will be given to the father of a child born out of wedlock only in very exceptional cases if the mother is not agreeing with it.
The German Supreme Court upheld this rule in 2003, so it is very unlikely that you would be given custody.
German Supreme Constitutional Court Decision 1 BvL 20/99 - 1 BvR 933/01
If the mother agrees the unwed father can join custody with her, which usually only is practical when both parents are living together.
It is absolutely unlikely that the father will be granted sole custody.
Nevertheless the father has some rights and obligations under German law. First, there is the obligation: he has to pay child support until the child is either 18 or completed its first vocational education, whatever is later, but never longer than to the child's 27th birthday.
Child support can be set by a court, which usually sets it at the rate suggested by the government. The income of both parents will influence the amount. The non-cash value of the care given by the parent the child is living with will be considered.
Usually child support rate for newborn children is between 200 Euros and 400 Europ per month, that is about $ 240-480 depending on the income of the parent.
The father in this case also has rights: the so called "right of interaction with the child" (Umgangsrecht). The father can meet with the child and do things together with the child. The law expects the parents to come to an agreement about how this right is being put into practice, how many days and how and where. If they cannot agree the parents can call upon the Family Court who will set a rule.
Only the parent who has custody can determine where the child will live.
A non-custodial parent who takes a child abroad against the wishes of the custodial parent will be charged with kidnapping. Germany and the U.S. have a treaty that they will enforce this in their countries and send the child back to his or her custodial parent.
Without being married to a German citizen it is not that easy to get an immigration visa to Germany, if you want to live there. You need to have an employment authorization and a firm job offer.