If you want to study at a Chinese university their academic year is different to the west. for example you have class Christmas day, have a look at this https://en.wikipedia.org/wiki/Academic_term#China It’s best you start studying at the start of the new academic year. But some universities will let you start mid year.
You will need to decide if you want to study for one semester or a full year, because you may only get a 6 month student visa if you say you want to study for a semester. Doing a full year will obviously be much better, in a sense it just comes down to “time on task” for improving your Mandarin.
Each universities accommodation situation is generally like this, they will have a dormitory for all their foreign students. Only the foreign students live here, the Chinese students will live close by but in a different building (go check out their accommodation 4-6 students a room) Foreign students get “VIP” treatment. Normally you will be boarding with another person, you may not have a choice who this is. Some universities will have single rooms, but they are hard to get. Another option is you may have to pay double rent to live by yourself in the foreign dormitory. Make sure you are 100% happy about the living situation inside the university before you decide
You can live outside the university, just by renting an apartment. You can imagine the pros and cons of this.
When you talk to the university check where all their foreign students are coming from, many universities have relationships with uni’s all around the world. But most have a focus, for example some will have more African students, students from Europe, America or Asia. 90%+ of the foreign students will be university students doing one semester or a year in China, their Mandarin levels vary! From fluent superstars to just learning pinyin. (For example most of the students in my uni were from South Korea, Japan and Eastern Europe, I was the only Australian)
Option A: If you need high speed internet,the best travel opportunities and money is not an issue live in Beijing or Shanghai go to one of the top universities, you pay your tuition upfront so you can go study anywhere, even the “top” universities like Peking.
It sounds like you have a strong work ethic and study routine, this is the really the most important thing!
A valuable topic would be “How to get the most out of studying in China”(because the real value I think is outside formal classes). I’ve got some tips about joining clubs, getting tutors, things to avoid. As will Gwilym and other people who have studied in China/Taiwan.
Sorry If I rambled on. If you want me to elaborate on any points further or have any other questions just ask.