From the east coast you probably should look (for the moment at least, prices are subject to change) at Eastern Europe as your starting point. We are almost certainly (75% sure) going to be starting our first RTW from Istanbul in late November or early December. So once you've figured out your route (and indeed your list is too long and not wholly possible with one xONEx) you could add on flights to and from your start/end point; if you fly on AA's codeshares with Turkish airlines to IST you would get q-points for those legs too.

Go to Fewmile's excellent oneworld site and download the rules and study them - many gotchas embedded there, especially regarding backtracking, numbers of segments and layovers, subsidiary airlines, blah blah. Use the great circle mapper and oneworld electronic timetable to see who goes where when. Become fluent in airport codes. All part of the fun.

I knocked together a play-time itin based on your list; it leaves out key destinations like Fiji, Bali and Thailand, plus short hops that in Europe are not only inefficient but use up valuable tix. I think this follows the rules but it's only for fun anyway. You don't need to send a check. Note that "/" means a surface segment thus CNS-MEL/SYD-HNL means you're traveling on land or separate ticket between Melbourne and Sydney. Remember, only two stopovers in the continent of origin. You need all six continents for your wishlist, which means fewer segments in each to stay under 20. Here it is:
Total q-miles is around 70,000, of which 60,000 or so, if in J/D on oneworld carriers, would earn 90,000 q-points. If the remaining 10,000 miles are at 1 pt/mi, there's your EXP.

You probably should get trip insurance and plan on a nice hospital stay when you're done. Given flight times, waits, non-daily service, etc., I would imagine this itin in a month gives you 1.2 days per city max.

Happy planning.