There is not "best" or "quickest" way. There is only one way. You go to the immigration office, have your Departure Card if you have one, 2 pass photos, TM30 (your registration of stay), if you live private, copy of passport of the owner + copy (or original) house book, and be ready to pay 1900 TBH. Best is - if you life in a privat place - if the owner comes with you.
No they are not. Thailand: one day quarantine, PCR test next day, self test 5 days later. Entering Thailand is only possible:
a) by air, rules see above
b) by sea, more complex rules
c) via land border if you life close to the border and work in Thailand
No idea about the other countries you mention. My Vietnamese friend here claims that Vietnam's borders are more or less completely closed. Did not check it, so :P
You can have an ED visa for any topic that is taught in Thailand, but obviously if you wold want to study Electric Engineering or something, they would raise their eye brows.
Typical topics beyond Thai as a language is Muay Thai (Thai Boxing), Thai Massage, cooking Thai food.
I would not wonder if classical dance and/or music instruments also are acceptable.