This is my DH's field so I have lots of experience in this area.
Yes, billing common area damages to all the residents is completely normal, expect a bill every year from now on.
Yes, they wait till the end of the semester or year to calculate and bill, would you prefer one weekly or monthly instead?
Usually it's divided up amongst everyone in the building and not just by floor - there isn't any reason why someone who lives on the 10th floor couldn't use a bathroom on the 3rd floor to dye their hair or for that matter a friend from another building. We lived in a 10 story dorm with 900 residents for several years, impossible to track down the culprit.
The trash was probably left in the hallways, elevator lobby, etc.
You would not believe some of the things that happen in these dorms - water fountains ripped off the walls, fire extinguishers sprayed all over the place, feces in the elevator or bathroom sinks, absolutely disgusting stuff. So much pee in the elevators. There are no cameras to catch who is doing it so everyone must pay the price. They used to post a running damage total in the elevator lobbies at one school so the kids could see what was happening and what they were being charged for, but it did very little to stop the destruction.