The GP is just a GP who is approved by certain hospitals to provide antenatal care. I'm not sure how they get approved, but they would have had a lot of experience with pregnant mums-to-be.
My GP wont be present at the birth of my baby, and I think it is pretty unusual for the GP to attend the birth.
As for the private room, it shouldn't make a difference if you are a private or a public patient. Being a private patient in a public hospital never guarantees you a private room. They probably try to give c/s patients priority with private rooms, however in other cases it is probably first in first served. The hospital I went to last time had just been refurbished and only had private rooms