Thanks for all the help thus far! In response to some of the comments/questions:

- My understanding of the rules is that even if I were to do BKK-(HAN)-SGN overland, that would still count as a segment. Did I misread that somewhere? Even still, I guess doing BKK-SGN overland would bring me down to only one extra segment. Otherwise, a cheap HKG-BKK-HKG seems like a good way to avoid the extra segments altogether.

By the way, in case anyone is wondering, there is an AKL-xSYD-BKK flight, but I was told I couldn't return to my point of origin (i.e. SYD).

- Yes, I'm doing LHR-xLAX-SFO on AA for the AAdvantage credit.

- I appreciate the additional ideas for maximizing my 20 segment allowance!

- I called the AA RTW Desk again today to inquire about "purchasing tickets in Australia" and was told that both the credit card billing address and physical mailing address for paper tickets have to be in Australia. I appreciate the suggestion to start in AKL rather than SYD (I would actually prefer to do this as it might fix some of my routing issues), but unfortunately I don't have any friends on the ground in NZ who can help me.