The health check is normally arranged by the recruiting outfit or school AFTER you arrive in
China.

Incorrect. The health check has to be done nowadays before arrival in China as they need this as one of the documents to provide you with an invitation letter. If they do it after arrival then they are not providing you with a z visa.

Did you research the name of the school or
recruiter?

Absolutely. I did not use a recruiter. I spoke to three of the fifty seven foreign teachers already at the school. I was interviewed by the Vice Principal who is Canadian. The only Chinese person I have spoke to so far has been a HR assistant.

Overnight
stays by guests depends usually on your relationship with the head of the faculty

Perhaps, I mean of course you would know. I am sure you are well intentioned with all your red flag stuff but if I was a HR person on the receiving end of your multitude of stuff I would simply file your application in file 13, aka ' the bin '.

Further correspondence with the teachers there have said that because a lot of the foreigners are married with families, they have tried to discourage overnight guests who are not immediate family within the on site dedicated accommodation. Apparently this rule does not apply to those living off site but this is few and far between as it is at your own expense.