Welcome to FT,
ROC!
Flying cattle class unfortunately (was thinking of booking business class, but they didn't have three spare seats all next to each other from what I could see).
Even if there are no three seats together, NZ will try to accommodate you at check-in. And if this doesn't work it is very likely that some pax would move to help you sit together.
I only once had passengers being uncooperative so they were then sitting between me, the baby and the Mrs DUF.
Re: the lounge you advised I might be able to pay for. I Googled "Air New Zealand lounge Terminal 2 LAX" and got a few hits. Is it the Koru Club one?
Yes, Kory Club is the name for the NZ owned lounges.
Can I book access to this lounge in advance, or is it a case of paying on the day and hoping there is room?
Only at the door.
One other thing you should consider: Unless you fly NZ a lot, Airpoints is one of the worst frequent flyer programs out there. You might want to reconsider using this program for this flight (unless your ticket is one which doesn't give miles on any other program).