Do this on Terminal:

defaults write com.apple.iTunes full-window -int -1

I'm wondering whether or not it's possible to put all ~400 users on each computer so that the grades can use each others' Macs interchangeably

Most certainly. I'm at a university where they do this on a much larger scale (think about 10–20k), so 400 should be possible — but, it also depends on the school's IT infrastructure. The user directories and settings are all on a central server (or perhaps server farm, I don't know) and each machine pulls the info when you login. You can login from any machine anywhere in the campus.

Typically, the maximum number of accounts the server can have is limited by the storage space, and the number of simultaneous logins are typically limited by the amount of memory, processing power available and network bandwidth (the clients need to communicate with the server and a single server can only process a certain number of requests at a time). So a definite answer is hard to provide.

The general approach is to spread the load (when demand spikes) over a cluster of servers or a server farm such that the end-user performance remains the same. Of course, no one expects the entire campus to be connecting all at once (there probably aren't enough machines for that). You can read more load balancing here.

This discussion mentions a software enforced default limit of 1024 simultaneous connections, extendable up to 2048. If you find yourself running at the far end of this limit, then you're either not balancing your load properly or you're under equipped.