Note that one a LONE4 you are not going to score a huge amount of status credits, so its unlikely you will progress from Bronze to Silver early in the trip, so we don't need to worry too much about the status bonus earned on BA verses the 25% base points. So I will avoid AA and BA as much as practical.

So how does this sound:

PER-ADL-AKL on QF
AKL-HKG-YVR on CX
YVR-ORD-ATL-DFW-BWI-DFW-JFK on AA
JFK-HEL-LHR-HEL-AMS-HEL-HKG on AY
HKG-NRT on CX
NRT-MEL-SYD-PER on QF

I make that 20 segments, 44,904 miles flown, 320 SC's and 37,978 base QF points.

note that if you could make one of the USA sectors via an alternate means (ie drive, purchase a cheap AA fare etc) you could finish USA with LAX-JFK-HEL using QF107 LAX-JFK earning more SC's and points.

ATL and BWI are the difficult ones since you need to go through ORD, DFW or MIA to get between them.

I also considered taking CX HKG-JFK (transit in YVR) and then continuing JFK-YVR on CX since it would be a connection from an international flight, but that added an extra sector and I could not then include ATL, BWI, ORD request and still get you out in AY from JFK.

You could leave from LAX/SFO on BA and get more SC's, but less points and more time in the air, so it depends on what you want.