Hi, there, thanks for posting this!
My main concern is that this seems more like a quick draft than a finished product. It feels like you're running down a checklist, not creating an engaging story or world for the reader. You're very light on details, and skip from topic to topic quickly, so none of them are really able to settle into the reader's mind. It feels like a how-to guide, and a very light one at that, rather than a fleshed out bit of writing.
The idea of creating a story based around a sorting system for the Library, has potential, but I'm not sure this is taking the correct approach. It feels, to be honest, too straightforward and grounded. The Library is a vast, mythic, space beyond the comprehension of most beings, and just a few genre tags isn't going to be enough to make something comprehensible. It feels too safe, too close to already existing classification systems. There's a lot of potential to go wild with ideas here, but you're not really using it. How would you classify something like a book that can only be read through dreams, or a movie that changes to match the life of every person watching it? You have the "can't be classified" category, and that kind of defeats the whole point. MOST of the stuff in the Library can't be classified, it's too bizarre, too unique. Your goal with this story should be to figure out how to attempt to sort those things into a comprehensible system, not throw anything not ordinary into a miscellaneous bin.
I hope this helps! I'm looking forward to see where this can go with editing.