How do I best fight for custody once my child is born?

Proving someone an unfit Mother can be complicated and must be done by a standard known as "clear and convincing evidence."  Although having a job or supporting the child can generally be a factor that the Court can consider, you seem to have many other factors that are in your favor:  you have housing for you and the baby and the support of your parents.  And you seem to have made efforts to find a job  but everyone is aware that the economy is not the best now and times are tough.  Keep trying.  It is the effort that will count.

Further, you have said that the baby's Father originally wanted nothing to do with the baby but now has taken a different position.  If he files against you in some way you need to get an attorney quickly to bring an action to establish paternity and demand child support.  You may want to go and get some advice now and be the first person to askthe court for help.  It may put you in a better place legally.

 He doesn't seem in any position at this point in his academic career to be making such a stink but one can never account for how people act.  Also make sure that it is determined that you are the custodial parent, regardless of the type of custody and visitation is determined between you.  Again, the home and loving family will be in your corner.

The baby is not even born yet.  You need to take care of yourself and the baby and not worry so much right now.  That is why seeking advice from an expert in your area may be something you should do now to put your mind at rest. Good luck.