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.

Not anymore.

If you are applying for a z visa outside of China then your school will have had to obtain you an invitation letter. This requires the results of a health check exam to be uploaded electronically to SAFEA.

This procedure then has to be redone when you arrive in China to satisfy a Government requirement that it has been carried out by Chinese doctors.

Our school follows a similar stance, they will not pay for the health check when it is being done in your own or a third country but will for the China side of things. I don't regard that as nickel and diming per se.

a seven page contract is really excessive

Ours is eight pages and I am at a top school in Beijing. A recent one I saw from a top international school was over 10 pages including an appendix. Not as unusual as you suggest.

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

It depends on where you are. Some schools, particularly Government ones and ones in areas where the local powers at be consider morality a big thing can be strict.
We are not allowed overnight stayers unless they are immediate family if we are staying on site. Those who live in their own apartments have no problems so long as they register their guests with the PSB.

And be advised from a law practitioner that your Chinese language contract will supersede your English contract

Probably. I have never had need to worry about it in eleven years. It is point of fact though that you will have to sign another contract which is either all in Chinese or a combination of Chinese and English. All provinces require this.