For us, our homebirth midwife will be taking care of the birth certificate paperwork. She fills it out and mails it in, and then we get a birth certificate in the mail from our state's vital statistics office. Normally the attendant has to file for the birth certificate because obviously that process is controlled.
As for social security numbers, either your birth attendant files for one when they apply for the birth certificate, or else you apply at the social security office after you have received your birth certificate.
http://www.ssa.gov/pubs/10023.html
For health care, I think our situation ends up being kind of more complicated than a hospital birth. Our midwife doesn't do the PKU, and neither does our pedi's office, so we have to go to the health department to have that done. Our midwife does a postpartum visit and we go to her office one week postpartum and six weeks postpartum. But our pedi still wants to see the baby for a newborn exam in the first couple days, so that's another office visit during flu season.

We're going to ask for either a first thing in the morning visit or else a clean room.

Anyway, that's our plan. Good luck with figuring things out.