Are you a dual national? Have you lived in the US for much of your life? Sometimes the ability to confer citizenship to your biological children may be limited if you haven’t spent part of your childhood (2 years?) in the US, plus a few more years as an adult, if they’re born outside the US.

If the birth certificate is an official one (usually with an imprinted seal), it should be fine. I believe you will have to apply for the passport through a US Embassy/Consulate in the country you are currently living in.