Call your health plan provider. It's different with each state, I believe.
Illinois you have to call insurance and let then know ahead of time you are pregnant I'm still under my dads insurance so baby can't be covered by it had to get the little one seperate insurance
In the state of Louisiana the hospital automatically enrolls your baby at the time the birth certificate is filed. Not sure how it is for your state though.
Yes your baby should be... I the state of Georgia you have 10 days after the birth to enroll your newborn.