1. You can get a great taste of Australia in 2 weeks doing the City, Rock, Reef combination (Sydney, Uluru, Cairns/Port Douglas). Of course, if your budget allows for more time, that's even better. Just don't try to cram too much into too short of a time. A good rule of thumb is 3 destinations for 2 weeks, 4 - 5 destinations for 3 weeks, etc. And remember that every change of destination is going to subtract at least 1/2 day of sightseeing from your itinerary.
2. Yes
3. As an American citizen, you will need to obtain the aforementioned Electronic Travel Authority (ETA) from the website of the Australian Government's Department of Immigration and Border Protection (www.border.gov.au) While the ETA itself is free, there is a AUD 20.00 processing fee.
4. Yes, those are options, but I would look at hostels in order to meet other like-minded travelers that you might want to sightsee and/or socialize with. Check www.HostelWorld.com for recommendations and reviews. You can also book through that website or directly with the hostel.
5. Most of the distances you are likely to travel will be too far to travel by car. So budget for a lot of air travel. Having a car is neither necessary nor recommended in large cities such as Sydney or Melbourne. Having a car in places such as Uluru and Cairns/Port Douglas is often recommended, but you can also visit these places without renting a car.
