Welcome to FT Jenneil

Looking at your itinerary, a couple of thoughts come mind:

• a LONE5 out of New Zealand is very good value in my opinion - NZD4099 plus surcharges & taxes (~AUD3300, ~USD2860) [an ex-NZ LONE5 is only NZD300 more than a LONE4 - excellent value IMHO]
• given that your are visiting 5 continents, I don't see an alternative that would come near to the value of an ex-NZ LONE5
• your taxes & fees seem high - at ~USD840 - this may of course be because you'll be ticketing with Qantas (no 'u') which is notorious for high fuel surcharges (using the on-line tool - in general the airline of the first flight does the ticketing)
• Perhaps instead - take your first flight on LAN - which happens to fly AKL-SYD - as it doesn't charge any fuel surcharges
• you're based in California and starting from New Zealand - have you thought of breaking the journey in California? Say after SWP/Asia/Europe and before Sth America?
• in my opinion you're wasting one of your 16 segments in flying LHR-CDG - take the Eurostar train and use the segment elsewhere - perhaps to go home to California before Sth America? Also since you have 'wiggle room' in Europe - why not drop LHR too - fly NRT-FCO and save another segment.
• with regard to your positioning flights - you could only need one - the flight from California to NZ at the start; you could 'forget' to take the final flight on the LONE5, the LAX-AKL flight. Alternatively, you can consider the cost of the second positioning flight, from NZ to home, as being the cost of having a second holiday in NZ one year after you started the journey
• if you're booking miles to AAdvantage, just be aware that JL only gets you 70% when flying in 'L', QF only gets you 50%, BA 25% and CX precisely 0%

Your current intinerary is: AKL-SYD-BKK-DEL-NRT-LHR-CDG,FCO-MAD-GIG-SCL-EZE-LIM-CCS-MEX-LAX-AKL
Suggested new itinerary is: AKL-SYD-BKK-DEL-NRT-FCO-JFK-LAX-MEX-MIA-GIG-SCL-EZE-CCS-LIM-LAX-AKL

• fly the first segment on LA to save on fuel surcharges
• fly these segments on AA (some are codeshare): FCO-JFK-LAX-MEX-MIA-GIG and LIM-LAX-AKL - a total of 24K+ miles; you'll easily make the Platinum challenge - provided you fly 10K miles in 3 months
• the second stop in LA has to be a transit (<24hrs) BUT if you're intending to forget to take the LAX-AKL flight...