When my baby is born will they say who the father is?

Depends on State. But usually your job, in addition to having the child, naming the father for the birth certificate.
Varies by state. Most states require you to name a father for the birth certificate. Then, in the majority of states, you can request testing through your local health department. This is done now by a swab on the child's cheek and one from potential fathers. This does require the dad to allow himself to be tested. Some states mandate potential dads to submit to test or be the responsible party.
Not usually. If there is a concern about paternity, depending on the state, you may need to have paternity testing done. In some states, that is considered a legal process where all parties (both adults and the child) have blood work done and the results can have legal bearing. Talk with an attorney for accurate information. Babies are not routinely tested in the hospital.
Only you. Know who the father is.If you are not sure then paternity test is in order.