[0] - I can't seem to find the correct word to describe the set of students across the years who pursue a degree in a particular subject (e.g. "Computer Science") on a specific faculty. I.e. the vertical slice.

Many in the US would call it the Department of Computer Science. At my university this was within the College of Engineering, Computer Science & Technology. There was a dozen or so departments in this college, Electrical Engineering, Mechanical Engineering, Civil Engineering, etc. There were other colleges like Arts & Letters, Health and Human Services, etc.