Korean won't matter, you can enter the postgraduate major after studying for about one year. . . If you take Korean students now, it is a good opportunity to learn Korean, but if you want to study specially, you'd better go to a language school here in Korea. .
Good English is of course helpful, because some professional courses here are in English, such as management department and some engineering majors. . . All the university professors here can speak English. If you don't want to learn Korean, you can choose a postgraduate major in English. .
You can learn Korean well here or teach Koreans Chinese (there are many Chinese courses offered in Korean primary and secondary schools). You teach Chinese as a foreign language, so I think you should choose education as your major. If you want to stay in China and teach Chinese, you should choose the Department of Chinese Education (hehe, there are also people studying Chinese education in our school). If you want to return to China, choose Korean education major.
Universities in Seoul, such as Hanyang University, Jianguo University and Yanshi University, are all good.
I hope my answer can help you.
I wish you success ~