Every imm has their own rules but it seems logical to do the TM30 again, because you have a new TM6 (Departure/arrival card) number, which is asked on the TM30 form
. There is no multi extension. When you have an extension of stay and you have to go out of Thailand, AND come back before that extension ends and want to keep the rest of the time from that extension alive, you need a single or multi re-entry permit.
, 30 days under consideration is very common for 60 days "Covid extension". Same for marriage extension. Maybe read some posts in Thai visa advice group ?
, a bit "visa/extension dependent" though. Y'r right concerning "if they found out living in Thailand on wrong visa". But the risk of refusing entry, is much much higher when you had Covid extensions (Unable to travel home). It is strange ofcourse when you leave the country and then come back after a couple of weeks. There are a lot of people refused entry when they had covid extension and then come back quickly...
, not 100% sure, but apparently you need a background check certificate from Thailand for a move to another country. You have to do this by all requirements mentioned in y'r picture.
That "letter from the Embassy........" is I think a letter for wat reason you need that Thai background check certificate. Because there are ofcourse no Thai Embassies in Thailand almost sure it has to be a letter from y'r own country Embassy (or government ?). Think they presume people are still in Thailand, requesting this.