when I was planning a natural birth and third stage, the midwife at the hospital wanted to write on the birth plan that we would wait 20 minutes and then I would have the injection. She stated that this was long enough for my "care provider" (I'm guessing she meant the obstetrician) to wait. She also said that as my careplan stated that we wanted to wait for the cord to stop pulsating before being cut, this would probably mean that by the time we'd waited for that, the placenta would be born anyway.
Well at least they were honest about why they want you to have the injection....that's a rarity in itself

As far as your questions go I'll have a crack

1. What, if any, is the relationship between the cord stopping pulsating and the birth of the placenta?

None that I'm aware of (but I'm not completely sure...someone else might know more). Waiting for the cord to stop pulsating is to give the baby the benefits and extra volume of the returning blood.

2. How is a placenta birthed naturally ie do you have someone pulling on it or let it come out on its own etc?

No one should ever pull on your cord if you're having a natural 3rd stage. The cord pulling is part of a managed third stage because the placenta needs to come out before the cervix starts to close (which is does reasonably quickly after the synto in comparison to a nat. 3rd stage). But pulling the cord increases the risk that bits of placenta will be left behind, that the cord will snap, and your risk of PPH. In a natural 3rd stage there isn't the urgency for the placenta to beat the closing of the cervix...hence no reason for pulling. You just wait and it comes out

3. What is the obstetrician's role (private hospital) in birthing the placenta. Obviously if you need stitches, they have to wait until the placenta is birthed, but what apart from that?

This is where we get to the crux of the matter. The obstetrician cannot leave you, nor can you be moved from birth suite, until the birth is complete. That means until the placenta is birthed - because if you are going to bleed this is when it is most likely to happen. Birthing the placenta can take up to two hours (or longer) without syntocinon (although most often it's out within 1/2 hour or so). Obviously the thought of sitting around for another two hours, when they can have it all washed up in 10 minutes with the injection...doesn't appeal. Not to mention tying up the birth suite.

HTH