Story list files are common ... they are generated (as I recall) when a user renames a story in the Assignments panel (like renaming a layer or a paragraph style). This doesn't change the name of the linked ICML story, it's more of a "cosmetic" feature to more easily locate certain stories in the panel within ID and IC, especially when ID is auto-generating story names during export. For example you might change a story's name in the Assignment panel from "05-AcmeLayoutABC-oneday" to "Headline".
There are other non-worrisome XML files that might be generated in the workflow, like if an InCopy user changes the order of stories in Story/Galley view by dragging their grey story bars up and down. Basically these files keep track of these sorts of changes so all users working on the project see the cosmetically-renamed stories in the panel/new order of stories in the Story/Galley view.
Nothing wrong with right-clicking to check out one at a time.
I've never heard of Story List files making stories invisible, but yes, you should be able to delete them without a problem. I would probably make sure no one has checked out any stories from the INDD file and that all stories are up to date, first, and then close the INDD file, before deleting them though.
Thanks so much!!! I deleted the story list files and now everything is fine. I still have no idea how it happened, we tried to recreate the situation with the editor, but of course now everything works perfectly. The publisher is very happy that it got fixed though, so that's the main thing