I wonder if this means SpaceX can build launch and landing facilities in Australia. Australia is really well positioned near the equator. I think it also has better weather than Florida, but I'm not sure.
Not really. The Australia Act 1986 just closed a technical loophole but Australia was for all intents and purposes fully sovereign since ratifying the Statute of Westminster in 1942. [1]
There's already a place in Australia I suspect is developed in anticipation