Note that if you start in Australia, you can't return LAX-SYD-AKL-Australia or USA-HNL-SYD-AKL-Australia. You cannot leave the country of origin twice. So your only option to hit NZ on the way back to Australia is LAX-AKL. This also means HNL is not an option for you since you can't back-track to the USA mainland after HNL and there is no flight option HNL-NZ.

But you can do NZ on the outbound journey. Something like Australia-AKL-HKG-Asia-Europe-USA-Australia or Australia-AKL-HKG-USA-Europe-ASIA-Australia.

Definately look to use AY for European sectors and use HEL as a hub (at least for their summer months). You can still go to/through LHR using QF or CX from Asia and then AY LHR-HEL. AY can also get you trans-Atlantic into JFK.