my kids are at year eight at patana and liking it despite changing over from the US system to British. Also, are you talking about moving now or for the new year? Many schools fill up quickly and may not have spaces at this point in time for this year. You may want to reach out to the various schools you’re considering and ask them about applications and availability , etc..
It is hard enough narrowing down which school when you pick a city let alone leaving it open to the whole country. There are search engines out there to compare them. The good ones are ALL expensive but you get what you pay for (to some extent. I’ve heard there are expensive crappy ones… but no cheap great ones). I knew I wanted Bangkok cause I didn’t want to have to drive. Do you want American system? British? In English or with some other language (Thai or European or Asian languages). People say st least in Bangkok you need to live near the school which is def easier. I somewhat envy the kids in my building who are going to school right up the street instead of commuting across town on bts like my kids. But my kids are very happy at their school (twins year 8)
Sounds like at this point since you have a bank account here your best bet is to go to Vietnam or Laos. Apply for non o there. Wait til it is approved based on home bank account amounts (or I guess Thai ones). Then enter the country. Then when you have 30 days left go apply for extension. Your account will be seasoned enough. And you will be good to go. Above you clarified that you NOW understand that you do not need background check or to go to the us. So your question should now count as successfully answered.
I have to admit I don’t understand your problem with this question. I flew myself and my kids here in business class both in December and when we moved here in July. I spent a total of approximately $30 in taxes to fly in absolute luxury. The points were accumulated on money I would have already spent so the plane tickets were basically free. 
Best way to accumulate points is by getting a card with a big signup bonus. If you aren’t loyal to any specific airline then getting a chase card is probably your best bet as Amex isn’t taken much in Thailand though you can still use the points. I don’t know if you can recommend other groups but… there is a good group to ask specific points related questions