Primary Custody and Child Support

My 18 year old cousin had a two year old child with his 17 year old girlfriend (who are now seperated) and is paying child support; however, she rarely lets him see the child. To add to this she’s having another child that is his. When she told him about the unborn child she informed him that the doctor explained that the baby may have down syndrome. Fortunetly, my cousin decided to speak with the doctor and found that his exgirlfriend was lying to him and in fact his unborn child was very healthy. He thinks she may have said this just to get more money from him. In addition to this, his exgirlfriend does not have a job so I do not understand why she has custody of the child and why he has to pay child support if he should have primary custody of the child. Is there any way he can get primary custody of his 2 yr old and the unborn child when it is born and not have to pay child support? Any help would be helpful. Thank you, Aaron