Can I get a 60-day tourist visa from Australia and then extend it for 30 days multiple times a year?

Yes get the visa from Australia, go onto the Thai embassy site in Australia, you can fill out the application form and then send in your passport, they will send it back with your visa inside. Then you extend it in Thailand.

Someone asked a similar question recently about getting another tourist visa in Australia and a few Australians replied that there's no issues getting a 2nd tourist visa in a short amount of time. The 30-day extension is irrelevant to your question since you can always apply for this each time you enter as a tourist (visa exempt or tourist visa)