MileageMonkey is a 3rd party tool which needs route changes added to it manually. It does not include all routes and could have routes still listed which no longer exist.

Qantas does have a seasonal weekend only PER-AKL flight.

AA has a number of same flight number tag flights, but they often change the destination of the short haul sector. eg, AA has flights JFK-ZRH with a tag flight on MIA-JFK, however that tag changes to DFW-JFK at different times of the year. They also change aircraft for the short flight, so you'd have a 777 for LAX-LHR and a A320 for HNL-LAX.
Yes, same flight number 2+ sector flights will only count as 1 sector to the ticket. eg, QF11 SYD-JFK will count as 1 sector, even with a stop and aircraft change in LAX.

If you're trying to maximize First class sectors, you might need to look at some of these flights again.
Qantas for example only has first class on the A380s and they only sell the cabin as First on flights to DXB/LHR, LAX and DFW. The SYD-PER is either 737 or A330 and PER-AKL is A330. You'd be downgraded to business class. The QF 737 business class is similar to the AA domestic "first" seat, while most QF A330s (26 of 28) have new business class suites.

The domestic AA sectors will also book into business class. After changes made a few months ago, AA 2 class domestic flights have the forward cabin sold as "First", but it is coded as business. The 3 class flights like JFK-LAX will be fine for First class earning though.