I think you’ve gotten lots of level headed responses. Having worked as a teacher (just an ESL - no teaching degree) I can verify that more often than not the wages are low ($900-1300usd per month) and you have a commitment 5 days a week, 8-10hrs a day with some extra duties thrown in. If you for whatever reason are disliked or disqualified from the job you lose your visa and work permit within days. You’re at their mercy so to speak. Without a bachelor’s your BF is out of luck work wise no matter how clever a “recruiter” tries to be. You mentioned volunteering but you still need the correct visa and work permit even if not being paid. Don’t spend the money on a TEFL unless you’re interested in teaching longish term otherwise it’s just a waste of money.
Boun Jannurak the US will recognize your Thai marriage but of course will need documentation. Probably best to get your Thai documents translated into English if you haven’t already. I’m almost positive that when visiting with her existing visa you can apply for her green card but best to check on visajourney.com which has a wealth of info. I brought my wife over on a fiance visa originally. Found that website to be a lot of help.