Some confusing replies here, apart from #5 and 2.
Traffic rights dictate that a 'foreign' airline may be able to operate for example HKG-MEL-SYD. It will be able to carry passengers HKG-SYD and HKG-MEL but not MEL-SYD, unless that segment forms part of an online connecting or stopover journey, in which the passenger will have been ticketed appropriately. The same applies on reverse on the outbound journey.
What they will not be able to do is sell tickets, in AU, to local pax, purely for the domestic leg.
There is an exception in the EU whereby the entire EU is considered for this purpose as one country, so for example an Irish airline can operate between MAD and ROM or even between ROM and MIL.