Some workarounds have been published here in the forum. They are based on copying the library files when the MC instances are inactive. I have explained my solution and Glynor has published nice scripts that can do the work automatically. However, these workarounds are not suitable for your not very computer literate dance teachers.
MC12 can use a shared library folder. The first opened MC instance has full read and write rights and the others load automatically a copy of the library on startup and cannot save changes. Even this is not suitable for your dance studio environment.
Another new feature in MC12 is Auto-Import. I think that separate libraries with the Auto-Import feature could be the best solution for you. The server would just provide a shared base folder for media files. Each client would automatically keep a local library up-to-date. When one of the MCs has ripped a new CD or edited tags the other MCs will automatically add the new files to the library or update the library from the changed file tags.
The users may need to use the same playlists on different PCs. MC can export and import m3u playlists so the users can periodically export all playlist to shared folders (The server could have separate shared folders for exported playlists - one for each PC). Auto-import should be able to import the playlists.
Edit
1. The teachers must be able to cut music ...
MC's own Media Editor might be good enough for basic audio editing like cut, paste, fade etc operations. If the tracks are ripped in a lossless format the audio quality will remain unaltered even if the same tracks are edited several times. I would recommend trying Monkey's Audio. The Fast compression option could be the best choice with Monkey's Audio because it would minimize the needed decoding and encoding time.