So much good advice, redtigeriii. Thanks!
To answer some questions:
-Business isn't required for long flights (which I was thinking as the flights across the oceans and from Thailand to Spain), but all 4 being on the same flight is. And, MSPeconomist, thanks for the tip about perhaps not being able to put the kids in coach.
-Thunderroad, I am willing to get more cards. That's the main way I'd plan on getting miles/points. We're paying what we can with cards, but our credit union holding our mortgage doesn't allow CC payments, and we miss a huge discount if we use a CC for insurance. We can make a 4k minimum spend in 90 days, but 5k can be challenging unless we have a big charge coming up.
-JDiver, thanks (I think) for the depressing AA news. As much as I don't want to hear that I shouldn't get my hopes up, better to hear it now. Our plan is currently to spend 2-3 months each at a language homestay program in Costa Rica, an apartment in Chiang Mai which will be our home base (my wife and I loved Chiang Mai), and in Spain where I'll be teaching a one-month summer class at the University of Seville (so nk15, that explains why Seville). What I was really hoping to learn was whether there are good sweet spots for miles in these areas to go on excursions for a few days. But your suggestion for camping in NZ is great; is there a company you'd recommend? And I'll certainly look into French Polynesia Moorea...thanks for the suggestion. Those seem like great experiences for when we leave Thailand but before going to Spain.
-Finally, nk15, thanks for the HH suggestion. That's the exact kind of tip I was hoping to get, because I wouldn't have thought of that program otherwise.