Is it possible to edit the view you get when accessing the file cabinet? I'd like to expose the file ID numbers in the view. Alternatively, is it possible to run a saved search on items in the file cabinet? (I couldn't find such an option, but I'm still fumbling around in here.)

You can run a saved search on documents in the file cabinet. On the upper right corner of the file cabinet page, click Search. Then you can click create Saved Search on the next screen.

Also if you enable the "Show Internal Ids" preference, the file ids will display in the File Cabinet