With as much travel in Australia and NZ as you're planning, you pretty much have to look at the Oneworld RTWs. Qantas is a Oneworld member, while the only Star Alliance member in the southwest Pacific is Air New Zealand, which doesn't offer service intra-Australia.
Oneworld has two RTW products, the Oneworld Explorer, sold on the basis of how many continents you touch, and the Global Explorer, which is priced according to how many miles you fly (and which also allows using some non-Oneworld Alliance members, but not very relevant in your case.) Of the two, the Oneworld Explorer is the better product overall – no mileage limitations, important for your itinerary.
However, both products have a lot of rules that need to be followed, one of which is that you're only allowed so many flights within specific regions (e.g. North America, Europe/Middle East, Asia, "Southwest Pacific" and so on.) The tickets allow a maximum of 16 segments including “surface” or “open jaw” segments (i.e. fly into Auckland, fly out of Wellington.) Within each continent, you’re allowed 4 flights, except in the case of North America, where you’re allowed 6.
Thus in the case of the SW Pacific region (Oz/NZ and some islands) you'd be allowed four flights within that area, and if you want more you'd have to pay for them out of pocket. But that might be for the best anyway - why burn expensive RTW segments on flights that could be purchased very cheaply?
So while your desired Australian/NZ itinerary might read like (USA) - AKL-WLG-MEL-AYQ-SYD-CNS-PER, that would be six segments, two too many, and in fact since there are no longer any direct flights between the USA and New Zealand operated by a Oneworld partner (Qantas dropped LAX-AKL a couple of years ago) to you'd need to land in Oz and cross the Tasman to get to NZ, using more segments.
Instead, you might do something like (USA) - MEL-AKL-BNE-SYD-PER, and visit Uluru (AYQ) from SYD and Cairns (CNS) from Brisbane (BNE) on separate tickets. You’d visit Wellington (WLG) as a side trip from Auckland. I know it's convoluted, but the rules of the RTW tickets are what they are, and if you're talking business class, the savings are simply too great to compare with over-the-counter purchases.
Now about prices. RTW tickets are priced VERY differently depending on where you buy the ticket and start/end the RTW. For example, a business class 4-continent Oneworld Explorer ticket purchased in the USA will be $10,799 plus taxes, fuel surcharges, and airport fees. The same ticket - with the same stops (except in a slightly different order) will be $6014 (plus taxes/fees) if purchased and commenced in Egypt, or $7900++ if purchased and commenced/ended in Copenhagen. HUGE difference.
RTW tickets are issued as e-tickets, so for example in the case of Cairo, all you would need to do is land at Cairo airport, go through immigration and customs, then head back to the departure desks, check in and fly off on the RTW. At the end of the RTW, you could end up in any Middle Eastern country - Jordan, Israel, Dubai... and fly home from there. (One of the many wrinkles in the rules is that RTWs commenced anywhere in the Middle East can end anywhere else in the Middle East, and RTWs commenced anywhere in Africa can end anywhere in Africa. Otherwise you have to end in the same country – not necessarily the same city – as where you started.)
Now obviously when doing the math, you need to factor in the "access" or "positioning" costs – the ticket needed to get to Copenhagen or Cairo, wherever – so it's not like you're saving all of that four grand in price difference. But can you fly to Europe or the Middle East for less than $4k round trip? Duh. Or, if you have FF miles, use those either before or after. Don't forget that you'll be earning a ton of FF miles in the course of the RTW, so the return trip home can easily be "on the house" using the FF miles you earned on the paid trip, so it's a twofer.
You mentioned possibly using FF miles to upgrade on long segments. You can only use FF miles to upgrade on flights actually operated by the airline with which you have your miles, and not in every case then. For paid RTW trips, it's really not practical - you buy a business class RTW and fly in business class on all the segments where it's offered. I know, hardship, but...
One other (big) advantage of buying Oneworld RTW tickets overseas rather than in North America (aside from price) is that the OW Explorer (continent-based) allows six flights within North America, so if you start somewhere overseas you can travel around and then fly “home” and use the six permitted flights, before returning to your starting point. Remember the tickets are good for a year, even if you want to do the bulk of your flying in a shorter timeframe.
For example, in your case, you could fly to someplace in Europe or the Middle East (where prices are way less than in the US) and start the RTW there. Then fly EAST to Australia/NZ and do your touring in the SW Pacific, followed by doing the Asia bits. Then you fly to the US and stop for however long you want. Use the North America segments to fly to anywhere you want – Central America (counted as part of North America) or the Caribbean, Mexico, Canada, Alaska… whatever you want. Use the flights for business or leisure. Note that business class RTWs book into first class on 2-class domestic flights.
Then, before the year is up, cross the Atlantic and finish the RTW in Europe or the Middle East. Use the miles you’ve earned to fly home, done.
Here’s a possible “backbone” RTW starting in Cairo that would meet most of your itinerary’s targets: CAI-LHR-SYD-AKL-MEL-BNE-PER-KUL-DPS-HKG-JFK-BGI-MIA-PHL-LHR-AMS-AMM. You start in Cairo, visit Uluru from Sydney, Wellington from Auckland, Cairns from Brisbane, and Singapore and Bangkok from Kuala Lumpur. You’d then cross the Pacific via Hong Kong (in Cathay Pacific’s amazing business class) back to the New York (then train back to Philly - don't burn an extra ticket.) Later you'd maybe take a trip to the Caribbean or the west coast (I picked Barbados but YMMV) and back, before crossing the Atlantic to visit Amsterdam, then ending the trip in Jordan.
You need to choose flights that are in Oneworld’s available routes. For example, to get from Perth to Bangkok you’d have to change planes in either Singapore or KL; if you pick KL you can visit Singapore as a very short (land based if you want) side trip and not “burn” the segments out of the RTW. Likewise, there are many low-cost carriers in SE Asia that can get you from KL or SIN to Bangkok cheaply, but save the longer trip (Bali - DPS) for one of the “scarce” RTW segments, in this case on Malaysian Airlines. Hope this makes sense.
It can all be quite dizzying, but if, after taxes, you’re spending something like $7500 for the ticket (ex-CAI) that works out to something like $470 per segment, including long, long segments on BA or Qantas or Cathay or Malaysian - all with incredible service. Believe me, as one who does one of these roughly every other year, it’s a blast.
I should also mention that you’ll easily attain high elite status with any Oneworld airline’s FF programs with just one of these trips. The route above would yield something like 100,000 miles in American’s AAdvantage program (if you know what you’re doing) – enough for another business class round trip from the US to Europe, South America, or Asia. Lots of bang for the buck.