As far as I know, Inner Mongolia is the only part of China insisting on police checks for foreign teachers. I could be wrong, but I can definitely say the vast majority of Chinese provinces don't require their foreign teachers to submit them. Re EF, don't get involved with them, or indeed their like. It's too risky to do so, there are many bad reports about them on the net. If you can't get a decent job direct, turn to a recruiter, but don't let him push you around.