I'm guessing it depends on whether his wife is Thai or not. My wife couldn't work on her dependent visa and had to get her own extension of stay based on employment, at least that's what her employer told her anyway.
the international schools I visited when looking for somewhere for my kids all wanted to see what visa my kids were on. They were my dependents so no issues but if on visa exempt/overstay then it might be a problem.
she won't be able to work on a visa exempt entry. More likely she'll have to apply for a new non-B outside of Thailand and hope the school pulls their finger out second time round.
speak to the hospital about getting a medical visa then. The IO is unlikely to give two hoots about your treatment when you've been living here on covid extensions.
Sure but be prepared to be quizzed over your covid extensions and not having the correct visa to live in Thailand, which is what immigration will most likely think you're doing.