Do you want to go to Darwin? If you do then you'll be doing a 'full circle' OR you could just travel the eastern states and then drive across to Perth. It's a beautiful trip if you like solitude.
You're budget seems about right but I wouldn't go for the corolla. Still stick with a four cylinder but maybe something jsut a bit bigger say a Camry or equivalent. (Stay away from Korean cars). That means a trailer with your gear inside OR a camper trailer (the Camry will hande a camper trailer without any probs).
Also, I am assuming you will be flying into the Eastern states from the US? ie. Melbourne, Sydney, or Brisbane. Arriving in Meblourne you can drive up the Easttern coast and then across to Darwin (it's a hot sticky trip and there's not much inbetween). OR you could arrive in Brisbane and drive south to Melbourne. That may probably be a beter idea given your Januray arrival. It can still be quite hot but if you are on the coast it will make it a little bearable. You can then take 3-6 months getting down to Adelaide which means the trip across to Perth is nice and cool across what is called the Nullarbour (sp?).
As for free camping ... I might leave that to Ozzy ... he's more up to date with that for the Eastern COast than I am.
Have you consdiered staying in Caravan Parks? It would work out to be about $180/week (max) for the three of you. That also gives you showers and a bit of security.
So far as insurance is concerned it might eb better for you to check that from a travel agency in the US ... I'm not sure how you will go trying to cover your car etc when you are not a permanent citizen.
The way I ahve written this is by excluding Darwin ... it's in the tropics but tehre are ways to work your journey to include it ... ie. Brisbane, Sydney, Melbourne, Adelaide, Alice Springs (Ayres Rock), Darwin, Broome, Canarvon, Perth. That covers just about everything.
The way I have it is Brisbane, Sydney, Melbourne, Adelaide, Perth.
So you'll have to decide which route you want to take.
Don;t forget this is VERY simplififed and until you know exactly what you want on your itinery we can't offer specific advise just yet. But the two routes I have suggested will cover most of Aus.