Yes business class will give lounge access. Having said that you probably would get access at AKL even if flying AKL-SYD in economy on the basis of arriving same day business LAX-AKL.

Yes US to NZ/Oz very hard to get business class other than full fare (ie limited availability both award and paid discounted fare).

Returning from CHC - would depend on the rules of the FFP whose miles you are using. Note CHC-LAX is discontinued so would need to fly via AKL (and economy only for all domestic flights).

In 4 or 5 days not enough time to see more than 1 or 2 places. As to where, depends on your interests. Adventure adrenaline activities? Go to Queenstown or Taupo. Culture & geothermal activity? Rotorua. Lush rainforest? West coast of South Island. Beaches? Perhaps Bay of Islands (nicer beaches in Nelson but still a bit cold for swimming in November). Etc.