Don't take a dog if you're planning to ride around Australia. Just don't. Forget that idea. For starters, the advice above, regarding dogs in National Parks, is quite correct - some of them won't allow to bring a dog even if you promise to keep it in your car (they more or less assume you're lying). What's more keeping a dog penned in a bicycle trailer, as you pedal around in the scorching hot weather, will probably earn you a tense interview with the RSPCA, and quite possibly a fine or confiscation of your dog as that will likely be seen as animal abuse - this assuming that you have already (as noted above) begun the complex procedure of importing a dog into Australia in the first place. You need to have started that a couple of months ago at the latest for an October departure. And not just paperwork, but multiple veterinary visits as well. And let's not forget that it will be required to spend at least ten days in quarantine once you're in Australia before it is released into your care.
The distances in the outback are long and very challenging. Do not for a moment underestimate that, especially down the coast of Western Australia if circumnavigating the continent is your idea. I rode down that coast in the height of summer, with daily temperatures ranging between 46-50C and found myself having to carry as much as 23 litres of water on my bike. And you plan to tow a dog in these conditions? Or have it trot along beside you? For hundreds of miles? On blistering hot bitumen? How much water are you planning to carry for the dog? And food?
It sounds to me like you really need to re-think a lot of this.
Even if you're planning to tour by car, re-think the dog angle.
