I suggest you get a quote for a shared container, to compare it with the cost of the Movecube. The Movecube used to be a no-brainer because it was cheaper than a shared container, but that's certainly not always the case now.
If you're not sure what to bring and what to leave behind, try doing a 'pretend shop' at the websites of major Australian stores. Sites like TheGoodGuys for electrical, HarveyNorman for furniture (or IKEA if you like their stuff, but make sure there's going to be a branch where you are, as they aren't everywhere here). The thing that stunned me, when I moved countries, was how much it cost to replace pots and pans, crockery, cutlery, towels and linen -- all the small stuff. We tend to buy those things gradually and don't realise how many you need, and how much they cost, when you have to trek round the shops and buy them all at once.
The dog is going to be the most expensive cost of the lot! Doing it yourself won't save much and isn't worth the hassle and potential disasters, look at Pet Air or similar.
You'll save a ton of money by having family to stay with, so that's great.
Buying a car is expensive, and Australians don't generally lease their car, because leases here are a dreadful rip-off. Also don't get sucked into a car loan from a car dealer, because they're a rip off too! Probably best to compromise on a second-hand car to start with.