Of those legs only ADL-AKL-SYD is operated by Air NZ (unless go via AKL for example SYD-AKL-BNE).

You can do all of the legs on Qantas, but the passes I am familiar with would require flying to Australia on Qantas.

Airfares in NZ & Australia can be purchased as one-way half the price of round-trip. Look for a sale and each leg should be A$150 + taxes or less. (I'm assuming BNE, ADL and AKL are all stopovers.)