Little strange AA ticketed
Flight 2: Qantas 7264 JFK-AKL (AA 7264 on QF3)
Flight 3: Qantas 7264 AKL-SYD (AA 7264 on QF3)
Could have been ticket as JFK-SYD. The date line may have confused the agent. Or QF not offering D JFK-SYD, and only JFK-AKL and AKL-SYD as D class.
But may get more AA miles.

Will need a NZeTA, even though doing an airside transit-->NZeTA required from 1 October 2019 to visit or transit New Zealand

Will have lounge access everywhere based on class of service (business)

AA thread for RTW earnings-->RTW / Round the World ticket - earning AAdvantage Miles

A map from the Great Circle Mapper - Great Circle Mapper

With a little planning you could have had a (free) trip to Alaska or where ever 3-6-9-11 months after your return to USA. Allowed 6 USA flights

With luck you will not be going to AUK, Alaska
https://en.wikipedia.org/wiki/Alakanuk_Airport