Upvote:1
I'm going to jump in here quickly and say yes, absolutely. Many tourists purchase a vehicle - a campervan or other larger car with a built-in bed, usually - and it's a super easy process.
I don't know if I'd recommend buying a new car though. You'd probably be better off sourcing something that's relatively new but still used. It will cost you significantly less and you should be able to resell it for a similar price at the end of your trip.
As far as travelling from Adelaide to Cairns, there are a few different ways to go. I've done the trip around the coast, and some of the scenery and coastlines are just incredible.
That said, through the outback would be an amazing experience as well!
Adressing a couple of comments: Many people use the address of a friend or family member, and I've met people who've used their hostel or hotel address. I've personally done this when I bought a car in Tasmania without a permanent place of residence. Regarding insurance, I have to say I'm not 100% sure how it works, but I'm sure it's possible.