Question: My Australian friend says yes. I'm not so sure, considering the monarchy collects no taxes from England, much less a country in the commonwealth. What's the verdict? Can she say "you're fired" and Tony Abbot is out?

Answer: 
The Queen (or King) of England is a constitutional monarch, a ceremonial figure whose powers are strictly circumscribed by the constitutions of England (or other Commonwealth countries).

In THEORY, the "chain of command" goes from the queen, to the governor-general of the Commonwealth, to the Prime Minister of Australia. In actual fact, the queen and governor general will do what the Australian (or English) Parliaments tell them to do. 

So the queen does "hire and fire." But not of her own volition, but as a result of the constitutional process.

The short answer is "she does," (what she is told to do) but "she can't" (on her own).
1. The Queen Cannot sack the Prime Minister of Australia.
2. In 1975, (I was there) the Governor General did NOT sack the Prime Minister, he has no power to sack the Prime Minister.
3. What can happen is this. If the Parliament cannot get legislature through parliament, due to blockage of the other side. Again and again. And 'supply is blocked'... i.e. money to be allocated, Parliament comes to a standstill. The Prime Minister can approach the Governor General and request a 'Double Dissolution' Both the Upper and Lower houses are disbanded, and a new election for both upper and lower houses are called for. This is what occurred in 1975. Whitlam couldn't get legislature through. There was an election, and Malcolm Fraser and Liberals were voted in.