Can I add my daughter's father name on her birth certificate? Also we were married at the time of her birth.

Laws vary by state, but you will certainly need the marriage license, and probably the divorce papers just to show that you weren't divorced when you gave birth. Ordinarily you need the signature of the father. Call your local courthouse and ask.

If you were married at the time of birth, why wasn't his name on the birth certificate? In all US states the husband is the presumed father and would normally be listed,