I experienced this whole process and asked my teacher about it in detail.
First, X2 is effective for 180 days from the entry and I did not experience such procedure as @dda mentioned. The application for the residence permit is only for those who have X1, in which case you must go to police station (actually, the police station that controls immigrants - 出入境管理处, not any police station). However, X2 holders don't need it and can't get residence permit. The visa for the short-term (one-semester) language program is X2. There is no medical test for me.
Moreover, you can travel to other cities in China during the class. It depends on each university (or maybe each province) how strict the attendance rule is. In my case, the attendance rate must be above 70% and otherwise the university has the right to "expel" students. However, this is pretty lax and it is said that no one is actually expelled in my university, though some were warned about the low attendance rate. My classmates didn't reach the attendance rate, but were not expelled. The only student that my teacher keeps warning is a student who is going to enroll the undergraduate program (本科生) from the next semester, in which case the student has a chance to lose the right of enrollment.
Some of my classmates attended another university in the past, and they said there were not such strict attendance rules.
The university must report the attendance rate to the local government, and if you have low attendance rate, it is possible that your next visa would get rejected, or otherwise you have in trouble, though.
Many students in my class often traveled to other cities, for 1 to 2 weeks usually, and multiple times. Some students in my class attend as low as 30% but never got expelled. Also, many students don't attend class even though they don't travel, as these students came to China just for party.
I'm still not sure if it "breaks the law" to ignore the payment completely, but at least once you pay, you can travel to other cities.