You mention you are transiting between international flights, will you go through Customs and Immigration in Bangkok? If so, you can leave the airport, but as mentioned by others, you will then need a Thailand Pass, Insurance and Vaccination records and also as mentioned, time is your enemy.
I used a Thai pass issued before the changes on May 1st and thought there may have been some confusion at check in and on arrival, but no problem. Was able to cancel the ASQ hotel and associated transport and PCR test, no issues.
Had the same experience on our recent return from Australia, being vaccinated helped and the only delay was getting through Immigration, but expected because of the need to check all relevant documents etc, but still only about a 15 minute wait. Just under two hours from landing at Suvarnabhumi to checking into our hotel near Don Mueang airport was pretty good I thought.
yes it is confusing, but after dealing with Thai requirements for a while, you get used to it. I would say only translate the English documents needed.
I would say both 6.2 and 6.3 are needed, if married outside of Thailand, because of the statement at the bottom about translation. The documents at 6.1 are required if married in Thailand.