Almost all campgrounds in Australia is bike friendly, but you'll probably find its much more relaxing (and cheaper) to just bush-camp. There's also so many free campsites available, too many to list (and it is a continent after all). As for amenities, you'll find most most (?) small towns will have washing/shower facilities setup for truckers (as well as the roadhouses) and lots will be free. Or you just go to the local pool and use their facilities for $2-5.
As for great rides- just pick a direction. It's all great.
PS: can't help you with airline. Have you checked out the "on the road" section above and the discussion hubs for some ideas?