I'm not positive if every practice operates the same, but first, no the ped is not present at the birth. They come sometime during your stay at the hospital depending on what time you give birth. If your practice has many physicians, there is usually one ped from that practice who makes rounds at that hospital on any given day. So, if you give birth on a monday, then Dr. A will see you. If you give birth on a tuesday, Dr. B will see you. If your ped office doesn't have privelges at the hospital you birth at then there will be a "house ped" that the hospital provides to do the initial checks.
After my first son was born, we found a great dr who wasn't really pro choice on vaxes, but he did respect the parents wishes. He gave us his speech about how he thought it was best, but didn't pressure us and he cared for us very well. When my second son was born, another dr from that practice showed up at the hospital and gave me a hard time about the vaxes. With the hormones raging, I was really upset and thankful that this was my second baby and I wasn't going to be shaken.
I tell you this not to alarm you, but to prepare you. If you have a good relationship or a strong preference for one doctor for whatever reason, just be prepared to take with a grain of salt whatever might come your way from other doctors. Just know that you will never have to see that doc again if they do rub you the wrong way! If you do have strong preferences about vaxes/circ whatever, be sure that your dh or other family who is with you at the hospital is clear on your wishes so they can back you up. (I'm not even sure if these are your concerns...but just incase, I thought I'd share my experience.)