Can an X2 Student Visa extension transfer to residence permit?

I am a student on an X2 visa and would like to possibly extend my stay in China (so I will get a visa extension through my university). If I apply for and get my visa extension, will I then be able to get a residence permit? If I am in the country for longer, it would be better to be able to travel outside of China.

Also, is it okay to travel to Taiwan on an X2 visa without having to reapply for a new X2 visa?