I have been in Thailand for 10 years on various different visas. The whole time I have been a remote employee. The Thai government is not going to spy on you in your home to see which buttons you're pushing on your computer.
My claim has always been "I work for a company in the US", Nobody has ever question where I do that work.
"In case you don’t know, most banks around the world don’t provide bank statements for the current month. Statements are automated and are provided only for previous months."
What is meant here is go to the bank and get a current statement. The have some sort of special letter or notification.
What they don't mean is the bank statements that most banks around the world send every so often that are outdated by the time they are received.
You can sleep on the beach for free. You can pay someone $1000 to cook you a $100 steak while you sleep on the beach for free. A very common expression in Thailand it's all "Up to you".
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<- that's a pretty good site to find local prices. You can get cheaper if you want.
500/month walking around money, its ok. You can have a good time that you will remember the rest of your life. If you're some working stiff and this is your big once in a lifetime get-away.... I would say, go for it.
However, I think you maybe over budgeting on your 'mandatory'. I don't think you have the money for "island hoping" but you probably have the money for 'Island staying'.
Koh Samui is a large lsland, you can visit a couple of the smaller islands for the weekend. You could get monthly rental off the beach for much cheaper than $700/month. A motorcycle rental is $150/month.
You can do similar for Krabi. If you rent on the beach $$$$ if you rent 2 or 3km from the beach $$. If you have a Thai companion $ (because they know their country better than we do).
I use google voice. Works for most things but I have some issues with the more serious verification stuff. You would need to use your friend's phone number and they'll have to verify every 3-6 months.
I have pulled visas from the Thai Embassy Washing DC while in Thailand. I then exit and return. You can pull a visa locally or from the US. For me it was 1 week. But that is no guarantee of entry, they have been cracking down.
I would travel to a land crossing and cross by land. I think that would be less of a pain in the neck if you get denied.