You've got a few misunderstandings in the process.
You do not get a proper Thai Visa from inside Thailand. As a US passport holder, you can just show up here and you will be given a (free) 30-day entry stamp called a Visa Exemption. You can extend that at an immigration office for another 30 days for 1900THB, so you could show up and stay in the country for 60 days.
You may be able to make a few border runs (go to a border to another country and back in to "reset" your time with a new visa exemption), but these are the thing most frowned upon now. If you're gone a few days it's easier to do, and if you fly out and back in, it's even easier. But they will still start to warn you that you need a proper visa if you do it too often. People used to just do this for decades, but noiw it's pretty tough.
If you go to any Thai consulate anywhere, you can get a single-entry Tourist Visa (1000THB). This is a proper visa that Thai Immigration prefers you to have. It gives you 60 days in the Kingdom, which you can also extend for 30 more days at any immigration office for 1900THB, meaning each SETV can give you 90 days in the Kingdom.
It's still relatively easy to fly to a nearby country when your time is up, and apply for another SETV at a Thai Consulate. Some are stricter than others about various requirements. The general guideline is going 3 times to the same Consulate before they will warn you to find a different way or demonstrate financial proof of various sorts.
You can ONLY get the Multiple-entry Tourist Visa from a Thai consulate in your home country (You would have to apply for it at a Thai Consulate in the US). All other Thai consulates will only give you a single entry Tourist Visa. I have heard of people getting them outside of this guideline, but it seems to be rare.
You do have to bounce out-in of Thailand every 60 days with the METV, but they don't give you any hassle during the valid period of the proper visa. So it's not the same as an out-in border run to reset your Visa Exemption stamp.
Can't really answer your question about the bank proof, as it's not something that often is actually required. I don't know anyone who has ever been asked that when applying for a Tourist Visa (that I know of), and I've never provided any. For SETV at most places, if they do question it, you're technically required to show 20,000THB cash, but honestly, they rarely ask that of Americans.
If they DO require proof of funds when you're applying, you'd better have it, or you're going to have to try another Consulate.
My understanding is that if you are there, you can get a 2 month visa (possibly called the Single Entry visa), then have to fly out and come back from an international airport, and you can get another month, then do an extension for one more, then fly again. Am I correct?
Get the single before you come, extend 30 more days for a total of 90, but then you need to bounce out and could come back on a VE stamp and extend that once as described above. That would essentially be 90 then out, 60 more then out and find some other way or another place if that's too much trouble for you.
If the school was too much time, there is another self-defense school that can help you get a Non-Immigrant ED Visa. I understand class is only on Saturday.
With an ED visa, you'll need to go to a local Immigration office every 90 days, but you don't have to travel outside the Kingdom until you need to renew the visa at the end of the year.
You could find a place to get a Non-Immigrant O visa for volunteer work at a qualifying place that will help you apply for that, and you only have to go to local immigration periodically to extend. Sometimes only as often as a year later. But you don't have to leave the country at all once you have the Visa itself. Ton of hoops to jump through, but if the place is legit, they will handle most of it for you.
Or marry a Thai person and stay as long as you like (foreign women don't even need to have 400,000THB in the bank to marry a Thai citizen and get a marriage visa like foreign men do )
Or be 50 years old and have 800,000THB in the bank and get a retirement visa.
No matter what you do when you are applying for any kind of tourist visa, be sure you tell them on the application and if they ask you: You are here for the purposes of tourism. You are not coming here to work.