90 day single entry visitor visa, how to change it?

ok here is my story: I actually intended to study in Beijing, where, upon arrival, I almost immediately had to leave due to a family emergency. So it wasn’t possible for me to register at university,get a student ID, and my single entry half-year visa also became invalid. So I decided to go to Taiwan so i could look for a University in Taipei.I got a 60 day visitor’s visa, found a university and with the letter of acceptance I went to Bangkok (meeting friends and doing visa at the same time, also I heard they were very easy with visas
) and applied for a student’s visa. Now, here it all begins. I now have a 90 day single entry visa, clearly marked as VISITOR. I also asked them to make it a multiple entry visa, but they wouldn’t do it. Is this actually the correct visa? Also
I am very unhappy about the fact that I am supposed to stick in Taiwan for the whole of 3 months and not be able to do some traveling. Can I go anywhere to change my visa status? I know these questions might sound naive, but I looked through the forum and couldn’t find something similar.
And how about going to China? How big is the hassle? A friend who studies in China even has a residence permit and can travel in and out as much he likes, and he got that pretty soon after his arrival. Yes, he is a student.
If this is possible in mainland China, it must be possible here, or not?
I am at Wenhua now and I am a German citizen, if that is of any help.

AND: I only go to Wenhua because I couldn’t find anything better on such short notice. Taiwan National University has a really good programme, where I want to apply soon, going to Wenhua is better than doing nothing at all.
Soo, if i want to change schools during my stay, how does that affect my visa?
I know, alot of questions here, but maybe there is someone out there who can help. Pleeease?