Congrats on your pregnancy. I hope everything going well with u and your little one inside.
As you are overseas student, the cost of the antenatal and postnatal care would be depends on your insurance cover. U should make it clear with your insurance company. Or, you may ring up to the hospital admission office which maybe able to advise you about the cost of the care.
To answer your questions,
1, 2, 3)
You should first go to a GP for blood test and urine test to confirm about pregnancy. Then ask the GP to write you a referral letter to the hospital or your private obstetrician.
As you are now 16 weeks pregnant, I think you should immediately go to a GP for first antenatal check up e.g. blood test, blood pressure check, advise on antenatal vitamin and nutritions.
In general, if you were PR or Aust citizen, it will be free of charge to use of public hospital services. Again, as u are overseas student, the cost of the care would be depends on your insurance cover.
4) I am from Sydney, so I am not able to answer this question.
Now, i went to two public hospital, they said they dont receive the person no medicare care, so my GP still perfer i go to public one.........it's save money.
I think you will need to pay full cost and claim it back from your insurance company. The cost may be quite expensive as well.
Have you ever consider to go back to your home country and have the baby there? Coz your baby will not be able to get the Australian citizenship if both you and your partner are not PR/citizen