The health check is normally arranged by the recruiting outfit or school AFTER you arrive in China. Ask if they'll reimburse you for the expense. Funny a big named tier 2 city international school is nickel and diming you over a health check. Convinced your well known international school is integrous? Did you research the name of the school or recruiter? And be advised from a law practitioner that your Chinese language contract will supersede your English contract and a seven page contract is really excessive. Also sending a contract back for revision is a normal process in negotiations. Overnight stays by guests depends usually on your relationship with the head of the faculty. Make friends first and ask for favors later. Sounds like you have all the answers. Good luck.