And do you guys have a special name for the $2 coins, like Canada calls then twoonies?
Not really, no. Australians are normally good at spontaneously inventing and propagating slang, and had nicknames for all the old predecimal coins (eg. trey, zac, deener and "two bob" for the 3d, 6d, 1/- and 2/- respectively), but none of their decimal replacements have ever acquired a nickname; coin collectors call the regular-design $1 coin the "mob of roos", but nobody else calls them that. We just call them "one dollar coins" and "two dollar coins".
The non-commemorative design of our $2 coin features an Aboriginal elder, and a native grasstree whose common name is also a racist slur; I'd reckon any nickname assigned to the $2 coin would either end up being a racist slur, or would end up becoming one.
We'd certainly never call them "twoonies"; that's a peculiarly Canadian invention, derived as it is from "loonie" becoming the slang for the $1 coin as it depicts a loon bird.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis