Another update: Downloading via Calibre content server does not work either. From some reason (may both feature or bug), the files are uploaded to the server with _number suffix, where number is the internal Calibre unique number of a document (Calibre Quick Start Guide gets number 1 and every added document a number increased by 1).
This means that they show up all as different instances in Kindle, creating the same problems as in case of delivery via Amazon. So either a Calibre content server modification or a separate server would be needed for this to work properly.
Oh well, I am getting too tired of this issue now and finally giving up.