Can't answer all your post, but will try on some.
In general, private hospitals are smaller, have more money, for privately insured patients. Much of it is elective surgery. Public hospitals are large, tertiary/teaching hospitals. They can have public (Medicare) or private patients. They generally have the most emergent services and sickest patients, most government resources. eg. if you worked in ICU, you would want a public hospital for sicker patients. Generally busier in terms of nursing. Pay rates for both public and private are awards, which you could google.
There are many agencies once you have registration who would love to have you.
Place to stay - check
www.domain.com.au which has all the properties you could need, including furnished apartments you could stay in until you find out where you will work, where you'll want to live etc. PM me if you'd like more info on this as I live in Sydney.
Very brave of you to move without visiting - but I hope you'll love it! Where in the US are you moving from?