Welcome aboard the RoadTrip America Forums!
There are a number of discussions on these forums that list all the problems and costs involved in short term purchases of vehicles by foreign nationals in the US. Suffice it to say that it is neither cheap nor easy. But to answer your specific questions...
The equivalent of the RACQ in the US is AAA, the American Automobile Association. They do maintain a list of recommended auto mechanics who could inspect your car, but they tend to be higher end (read costly) garages. You can get a basic road-worthiness inspection at almost any garage, so I would suggest you ask your girlfriend's family for a local recommendation. But to be blunt, I doubt that any van you can purchase for under $2,000 would be up to an extended RoadTrip.
Having a local address will help in registering and licensing any vehicle you do purchase, but it must be the address of the owner. If you and your girlfriend both have Australian citizenship and Australian driver's licenses, then having a local address won't help all that much unless her family would want to take on the responsibility and liability of being the owners of record and then giving you free rein to do as you will with it.
As for insurance, your record helps, but the biggest drawback you have is your age. Auto insurance for those under 25 is expensive for the simple reason that drivers of that age group have the most, and most expensive, accidents. That's how insurance works. You pool the risk and spread the cost. You're in a high-risk group, thus the higher cost. Again, you might be able to piggy-back on your girlfriend's family's insurance and get a multiple driver discount, but that would be a discount from the higher age-related rate, and if you then have an accident, it would cost her family for years to come in higher rates.
In short, there are no easy or cheap answers. If your only use for this vehicle is a relatively short-term RoadTrip, then renting rather than buying is the cost effective option.
AZBuck