When you download a book through the Overdrive App, it is only useable/readable in the OverDrive App. Once a book is downloaded and you go to Overdrive's "Bookshelf" the cover for the book is indeed shown.
This doesn't solve the problem of the file icon on the device but, in my opinion, is a unnecessary as you don't really need to access the file directly from the Nook menu as you can only open it with the OverDrive app.
I don't use Calibre myself so I'll let someone else comment on how to use it, but I don't think it will help you with libary downloads anyways.
Thank u so much!