some thoughts:

it is not 100% clear where you are transiting and where you care to stopover so pardon some assumptions...

for Jordan, a week is better, you can include Jerusalem by taxi which shouldn't be missed if you've never been.

amm-hkg direct is better as pointed out already unless you are visiting BKK

mnl: sorry no experience in Philippines

dfw-ewr-mia-scl: do you have to go to ewr? mia? you could do dfw-jfk-scl on AA/LA

along same lines, for Arizona, LAX is better jumping off point; low fares available or you can even drive. so you could do hnl-lax-scl if you didn't need ewr or mia.

scl-puq-scl: you can buy these flights for about $120 ea. if you want to leave them out of the RTW and save segments, especially since they are only Economy anyway. 2 days down there CAN'T be enough!! (never been but just a hunch; same hunch with Easter Is.)

as far as I know, any OW carrier can book the tickets, people say AA has the best RTW reservations desk. i think now you can even book i all online??

enjoy!!!