Hi,
Yes there is a private and govenment health system. You need to purchase your own private health insurance policy and there are alot of different health funds to go through. Normally, you have to be covered by a health fund for over 12 months before they will cover obstetrics. You do have another option and that is to be a self funded patient in which you pay for everything yourself
In regards to whether you get a choice in a vaginal birth or c/section and how many ultrasound you have is completely dependant on who you go through for your ante-natal care. I went private and my obstetric provider did an ultrasound ever visit and he always started off with the goal of vaginal birth but he was completley open to a c/section if there was a valid reason behind it. I think a 'normal' schedule for ultrasound of a normal pregnancy in the government system is one at around 12 weeks and one at around 20 weeks, I could be wrong through.
Best of luck with your move.