Depends where you're traveling too, each province has different entry requirements, and what your vaccination statues is.
Airlines themselves don't require a PCR test if you're vaccinated. Some of the islands require one to enter, Ubon Ratchatani requires one, not sure where else.
Work permits are issued same day if the paperworks in order. If you haven't gotten your work permit, either your employer hasn't submitted the paperwork or they are not eligible to employ you.
You should note that working in Thailand without a permit permit is illegal. Some other countries like Vietnam allow foriegners to start working while the WP is being processed. That's NOT the case in Thailand, you cannot start working until you receive your workpermit.