The following airlines fly from LAX to PPT (Papeete):

Air New Zealand
Air France
Air Tahiti Nui
AOM

I think that's it. Corsair flies from Oakland (it's a flight that originates in Paris)

Hawaiian flies from Honolulu.

As far as partners, Air Tahiti Nui has no code-share or mileage partners right now, as far as I know. But it does have the best livery of any airline in the world.

Air New Zealand is a Star Alliance partner so you could use United, Lufthansa, Thai, Singapore, etc. miles. I believe on United it is 60,000 in coach, 90,000 in business and 120,000 in first. I flew ANZ from LAX to PPT a couple of years ago in coach and was not altogether impressed. But it is better than AOM (which is about to go out of business) or Corsair (which has tiny seats from what I've heard).