1) Not much F to NZ. LA has SYD-AKL-SCL. Otherwise on OW you'd need to go via SYD or MEL (with business class for the trans-tasman hop).

IIRC AA miles can also be redeemed on FJ and TN, which both fly LAX-AKL via NAN and PPT respectively.

Awards in premium cabins are difficult to get. When are you intending to visit.

2) Time of the year makes a big difference for NZ beach activities. December to March being best, but also busy travel period.

3) Lots on offer in NZ. Check out the NZ master thread (sticky in this forum) for some ideas.

4) NZ generally cheaper to visit than Australia, in terms of accommodation, within NZ travel, food/drink costs. Activities could be cheaper or more expensive depending on what you want to do.