Can I enter Thailand on a visa exemption and then extend my stay after returning from Australia?

should have no issue comin' in visa exempt for the first visit, then returning visa exempt for the second one too. You'll have to take a 30 day extension at the immigration office in your area for 1900baht but that shouldn't be a problem either.

I can't see the need to get a tourist visa from Oz before you come here unless you don't wanna fool with the in country extension