Australia, New Zealand, the UK, etc. have standardized rates for stipends - and in fact for minimum wages in general - that's also inflation-corrected and increased accordingly every year. Though the minimum wage stuff also is there in the USA, there is no inflation correction in any government jobs/public (and most of the times private) universities. The stipend you have mentioned is calculated from a standard Australian government agency that governs (indirectly) the APA awards. Hence I would not worry about its calculation as it would be correctly calculated to the precision of the inflation-correction till this year. I used to get around 18,500, 18,800 and 19,150AUD (approximate figures) back in 2007 to 2009 in Perth, and my wife and I had a happy life (we didn't have kids then though)! I could afford to rent a 1living room+1br+kitchen apartment (called unit in Australia) some 7kms from the city center. If you want to live quite close to the campus, then you will have to pay quite a bit - or share an apartment with others. I could buy and drive a used car. There is not too bad (much better than any city the USA except NYC, for example, but not so great compared to Europe) public transportation. The tickets and monthly passes were discounted for the students.
My wife and I went back to our home country just once in 3 years but that's not because we couldn't afford it but for other personal reasons.
One more thing - you don't pay tax on this stipend. So you get the full in your pocket.
You obviously can't afford too much partying, eating out, etc. with this money. But for a student with usual studentsy life-style, the stipend is calculated well. Or, the other way around, if someone is spending more money than this stipend, then he\she is living a bit more expensive life than the 'average' student - not passing any judgement here, just giving a practical financial comparison.
The health insurance was covered by my award for me and my wife as well. I don't know if and why they have changed the rules recently about covering the health insurance.
For the other paid work such as tutoring. It may very well be even more than 20AUD/hour. The academia is (mostly) unionized there and the union negotiates the wage-range for each job in academia including tutoring every year to make sure that the inflation-correction is correctly implemented in the wages.
I was also an international ph.d. student in Australia, and working in the USA these days. And I am ready to say on record that Australia has been the best thing that has happened to me - and I would love to go back for a permanent job there if there is any such opportunity! I really wish you get the time of your life both academically and personally.