They actually are. When the government needs help they always flock to Qantas. It's majority Australian owned, and believe it or not they actually make money.
Not in the true sense of a national carrier. There is no ownership. Military transport is always done by some sub-contracted charter outfit. Many other operators charter for the Australian government too. Alliance goes flat out during an election. I would hazard a guess that the political blowback from not using Australian Carriers in the last 3 months were why they went with QF. China Southern would have been cheaper but they wouldn't want the publicity. Qantas Pilots likes to think of it as a National Carrier but it isn't by any stretch of anyone's imagination.