No, you don't need a birth certificate. If you have an old passport, even an expired one, they accept that as both ID and proof of citizenship. I just did this, and they needed the court order, the doctor's letter, the completed DS-11, a passport photo, and my long-expired passport. On Tuesday morning I went to the passport agency, the agent put the letter and the court order in a Priority mail envelope and clipped the old passport and the form to the outside of it, and I paid him the fees (including the $60 expedite fee). I received the new passport and my documents in the mail that Friday!
Give or take a two-hour wait sitting on an uncomfortable federal chair, it couldn't have been easier, and the $60 fee was well worth it.
To get a 10-year passport, you need to make sure that your doctor uses the version of the letter which states that you've completed transition.