Here's what I would do to get to each destinations you stated with the miles that you currently have in your account and the future addition of US airway miles.
South America:
Search BA.com for award seat availabiltiy on when you want to fly in 2012. You can fly from JFK-LIM-SCL-IPC-SCL-JFK or some crazy itinerary look at the BA thread for ideas. The nice thing about BA miles is that you're allowed unlimited stop overs on the way to your destination, so you can see more than on country on your way to your destination. Once you get your itinerary set, transfer the Amex MR points and Sapphire points to your BA account. Biz class to South America is 80k Round trip per person, so you have more than enough miles for you and your GF.
South East Asia/Europe:
I would use your US airways miles to get to South EAst asia because star alliance has a better presence there with Singapore airlines, Asiana, and Thai airways. With that said, it's 80k in economy or 120k in business. The best part about US airways miles is that you're allowed one stop over. So you can visit Europe on a stop over to South East Asia on the same award. So you're killing two birds with one stone.
Europe/Hawaii:
Using your AA miles, you can use the AA partner award chart to book two one way award tickets. One ticket from New York to Europe and another from Europe to New York using New York as a stop over then onward to Hawaii. You're stop over in New York can be as long as 12 months. Then you got to figure out how to get back from Hawaii to North America.
You can use your US airways miles to get back from Hawaii to North America using Norther America as a stop over and then proceed to South East Asia since the redemption rates are the same for travel from North America to South East Asia as it would be for Hawaii to South East Asia.
There are plenty of options to get to your destinations, it all depends on what's available when you want to travel. You can make this happen by maximizing stop overs.
If you have any other questions, just ask