The G-G dissolves parliament every time the Prime Minister (PM) advises them to do so. I think you don’t grok the situation here, constitutionally speaking.

1. The King (or Queen) of Australia has powers defined in our constitution. They can’t issue commands at will.
2. The King appoints the Governor-General (GG) on the advice of the PM
3. The King delegates their powers to the GG
4. The GG acts on the advice of the PM, to approve legislation (royal assent), and to dissolve parliament when the time comes. Also, awarding honors and some other non-political stuff. Head of state duties like greeting and hosting other heads of state.
5. The GG does not seek permission or even advice from the King. Delegation of powers doesn’t mean the GG may exercise those powers, it means they must exercise those powers. That’s an important difference.
6. There are reserve powers, “break glass only in emergency” powers. One of those is to sack the government. It’s happened once in living memory, in 1975, when the elected government couldn’t pass funding bills and the government was about to run out of money (sound familiar?). That’s one of the few triggers where the reserve powers can be used. They can’t be used for just anything. Sacking the government also means a full election, upper and lower house.
7. The GG doesn’t report to the crown (King or Queen) in the sense you mean. There’s no “list of things I did today” and the King then sends back an “approved” stamp.

He dissolved parliament based on what rules written by whom, on whose orders?

Hint hint: Based on the Australian constitution, written by Australians, on the order (well, “advice”, same thing in this case) of the Australian Prime Minister.