Singapore is probably the most expensive destination you could choose for a short flight out of Thailand .
Do some research on Phnom Penh Cambodia as that's often a short cheap flight from Bangkok..
You will need a visa to enter Cambodia but visa on arrival is available for most passport countries or you can apply online say a week or so before you travel and save the hassle of lining up at the visa on arrival immigration desk..
Phnom Penh has quite a good nightlife and bar scene and it's relatively cheap for a short stay..
Vietnam could also be an option but depending on your passport country you may require a visa before you enter Vietnam (British passport holders for example don't require a visa for Vietnam and get 45 days entry on arrival) but many other countries require a visa prior to entering...
HCMC probably the best choice for a short visit and stay in District 1 near Bui Vien walking street for some nightlife and plenty of hotel options.
You can declare yourself a non resident for tax purposes.. your choice
But you would only do that in a situation where you have overseas income and paying tax in another country and don't want that income also subject to Australian taxation as well.
As long as you are leaving each time before your 60 days expires you will receive another 60 excemption entry each time you enter Thailand.
As long as you have proof of a flight to exit Thailand within 60 days it doesn't matter to the Thai immigration doesn't care where you are going as long as you are leaving the country.