It depends. If you want a real answer you'll need to provide more details.

All the jobs have been filled by Eastern Europeans, so tough luck for you.

No, why?

Now, easy, April 1st 2017. No, not easy any more unless you're top talent. Can you say A, B, C ?

http://shanghaiist.com/2016/09/12/work_permit_ranking.php

Depends on your race, age, gender and nationality.  Sure glad the Chinese are not biased

Sure, you probably could. 

But depending on your qualifications, nationality, race... etc... 

Then it will very well determine how much you get paid, living standards, how you are treated... and most likely, you will just be confined to smaller towns/cities. 

There is a plenty of teaching jobs for English speakers (not only for natives). Native speakers should have no problems finding a teaching job in some of the major cities. As for the non-natives it all depends from your luck, but many companies won't be willing to offer you a working visa if you are a non-native teacher. What is more, unfortunately Chinese people often don't want to hire black people or native English speakers of Asian origin. 

Why not click on the website link to the "Jobs" section, choose a few of the many teaching jobs advertised on there then apply for them? That will probably give you a much more accurate answer to your question.