It might vary by state. Here in NC my husband filled out the birth cert. at the county health department. He just walked in & said we had a homebirth w/ out having to show proof of pregnancy or even me or the baby. She just smiled & said homebirths were becoming more common (though illegal. I don't think she knew that.) That's the only registration you need for the birth itself, you'll probably need it for insurance.
We decided against a SSN, it's only for tax purposes & we don't pay taxes. She can register herself if she wants to when she's of age & wants her own job.
The hand & footprint is purely a memento. You can get your child printed at the police or sheriffs office if you're worried about kidnapping.
The pedi is optional unless you want to vax on schedule. I wanted the appt. w/out vaxing & they worked us in as soon as they could, a week after she was born. Things move slow here.
Doctors & nurses here were shocked & thought we were crazy but impressed after hearing our prenatal care set up & seeing the records we kept. Having a healthy baby to show, having a previous HB & Husband just finishing EMT training probably helped us too.
Getting the birth certificate registers the birth, there's absolutely nothing else you have to do. Getting a SS card is optional.
I would call your insurance before your baby is born and ask how to put him/her on it yourself after they're born. This way you're not waiting until the last minute to figure that out. Its probably fairly simple but it helps to know.
You can take your baby to the pedi at any time, there is no rule. My daughter was perfectly healthy so i knew there wasn't a real need to take her right away. She didnt need to go for several months, actually. The doctor found it odd, but didn't make a big deal about it. Our fears often make things like this seem worse than they often are. It is your baby and you are responsible for his/her health, not the doctors. If you feel confident with being able to notice any problems then there's no reason to rush to the doctor. If you don't then a quick checkup is good for peace of mind.
Above all, remember that birth is rarely a true emergency, enjoy the process as much as you can and love is what your baby needs the most.