Re teaching contracts in China,and clauses about taking on extra work with another employer: it used to be the case that many contracts forbade this. I think that was because the sponsoring employer didn't want other Chinese employers to utilize the services of his FT's! But some foreign jobseekers objected to such a clause in their contracts, and so rejected job offers from prospective employers who insisted on it.
But don't ever underestimate the Chinese. These days the offending clause has been replaced by one that says you must ask permission first. Actually, I can assure you that that permission won't be granted by many employers. That said, I think that's reasonable. From a tax aspect alone, that is perfectly acceptable, if for no other reason: two jobs will usually mean that the threshold for being ineligible for the payment of Chinese income tax will have been exceeded. Tax evasion is a serious matter!
Having said that, I should say that I know some private employers ask FT's to sign two contracts with varying terms and conditions. Never be so foolish as to sign twice, or else you may be a party to some kind of Chinese tax evasion!