In Orange County NY I called the courts while pregnant and was told that I could not file for child support until the baby was born. He will need to sign an acknowledgement of paternity at the hospital & if he refuses, the court will make him take a DNA test when you do go. But you do need to file immediately after the baby is born to get arrears. Also, remember not to go into court with the attitude that you want to be supported. He does not have any obligation to support you, only a partial obligation to support your child. The courts get angry when it seems that people are there for themselves & not the child.
Best of luck! Court is hell, but your baby is so worth your efforts.
I just wanted to add to my message after reading others:
I'm sure everything is different in Counties & especially states. In Orange County NY however, you can have whatever name you want on the birth certificate. My sister had an unusual situation & her children had a different last name than her & their father. You don't have any legal obligation to use any last name, honestly you could make one up if you wanted. My sister used her fiance's mother's maiden name for her children because of some issues & now all 4 of them have that last name, but her son is 2 1/2 & has always had it. The hospital can not tell you what to name your child.
Also, you do need to file for custody right away as well, that is so important to have.
I am only recieving about 10% of my son's father's income due to different circumstances, so nobody gets an exact amount, the courts will tell you when you go through it, it can be very frustrating though, what I get can barely buy diapers for the month... also if a man is on public assistance they do not have to pay anything! Is that insane or what? But it's true, I've seen it happen. So basically expect the least, and hope for the most.
& one last thing... the court clerk told me that the most important peice of paper to have is the acknowledgement of paternity, even more important than a birth certificate. They will give you one at the hospital for you & the father to sign, and then it gets filed with the state. I went through that because we were not married either.