Well...About your questions...I think the last one I can tell you something.
Universities in China...At large, expect those top ones like Tshinghua, Peking University. They need foreign teachers, so it's relatively easy for you to find a job there.
Usually Chinese students welcome foreign teachers especially from Europe and United States. It's better if the teacher is a female:)
If you want to find a teaching job, I suggest you pay a visit to those universities' website first. Locate the HR department, send an application, including your CV. Of course it would be super if you have teaching certification in your country.
At my experience, try to find some second-class universities first...you know that. There're few opportunities in top class universities of China...you know that...maybe. And another advantage of choosing second class universities is that you can gain experience in your CV when you want to jump into a better one. (But i saw you are studying in peking university, that's great...)
English studying institutions is also a good idea, usually with prolific payment and potential opportunities of getting in companies...you know their target customers are company employees or managers.
About Chinese improve...Well, I think you should know the term of "Immersion study". In such a great environment you don't have to worry about your Chinese, you got plenty of chances to practise. Cos you have to use and practise if you want to have a good life here...Like survive, huh?