1) Am I missing a "secret entrance" to the academy - one that doesn't require me to keep going through those preliminary rooms? I liked those rooms a lot - they were educational and helped establish the game's aesthetic - but after going through them a couple times, I'm kinda done with them, or at least would like to be.
Yes. From the entrance, there is a door that will take you into the cavern section.
2) Is it OK to quit inside the Academy?
I believe it's fine in any room. If you type "quit" and it works, it's fine! Otherwise, you can 'recall' anywhere in the Academy, so you're always no more than two commands away if you need to depart.