correct, they want qualified teachers with the necessary teaching qualifications. In short, they hire career teachers so if you want to teach in one then you will need to do the necessary qualifications and get some actual teaching experience.
as an example, Emirates were fine with my online Thai vaccination records (in English) when checking in at Heathrow for flight to BKK last month. No one asked for them at BKK, just passport and boarding pass.
they just want to see evidence you've been vaccinated. Whatever they use in China should be fine provided it's also in English. If in doubt give your airline a call as they'll be the ones who will decide if you get on the plane or not. I wouldn't worry about being checked at BKK, I'm not aware of anyone being asked to show it.
who knows, no one can predict whether an embassy will give you a visa or not based on your history. For the cash, if requested by an embassy then a statement, if requested at border/airport then cash.
not necessarily no. If you've beenl coming in and out of Thailand regularly this year then they might consider you to be living and/or working here under an incorrect visa. On the basis you requested been in Thailand for 5 months already this year I'm guessing this is your issue and a 2-3 week stint in Cambodia won't necessarily convince them you're a genuine tourist.
with a company with an entity in Thailand who will then help you through the process as they need to cut down several trees to give you all the paperwork required