Marry, take her back to the USA and apply for her VISA there. Don't do any VISA anything here if you can avoid it. My wife's tourist VISA lasts 5 or 10 years something ridiculous like that. In another words, get out of Thailand as quickly as you can. You can do paper work back in the States.
I guess children born here are Thai and if born in the USA they are Americans. Thais can't have dual citizenship. Anyway, where would they go to school? How would you provide for them? Any school here worth anything is about 1 million baht per year. If you're lucky you'll get 600-700,000 per year before taxes? The thought of having kids here makes me sweat a lot. I'll stop thinking about it now.
On a personal note, think about what you are doing CAREFULLY...LONG AND HARD before you make any decisions about coming here or marrying a Thai. Is there a gringa somewhere that bakes your noodle? Come on... try ... try! you can do it!
Does she know you have that Avatar? Seriously, WTF?
anyway, Good luck.