Are they removed from the Downloads folder in the Library?

Also note the the downloads are treated differently when you are in private browsing mode and are only shown temporarily on the about:downloads page (this page works in both modes).

Downloads would normally not be removed unless you would clear the downloads via the right-click context menu or otherwise clear (recent) history.

The files are still there, but the list of Downloads (which I often use to locate the downloaded files or to revisit the download page without too much searching) emptied in front of my eyes. I started downloading two files, pulled up the Downloads window (about:Downloads) to check the speed, saw the full list, and then a moment later only the two files currently downloading remained. I didn't click anything to clear the downloads, and nor was in a Private Browsing session.

Your point about downloads only being removed by a user action is what I thought was the case; but in this case, the data disappeared without any action on my part.

Frustratingly, Firefox doesn't make a backup of that particular file, but I've read elsewhere online that perhaps the data is still there in the sqlite file, just marked as deleted? Is there any hope for recovering 12 months worth of download links?

The places.sqlite file isn't replaced by a new copy, but is opened for reading and writing and records are updated and removed, so the current file is only modified. So there aren't any copies of this file available for backup. It is however possible that the records haven't been removed yet, but are marked as deleted. You would have to save a snapshot of the current version of the file to have a backup to undo changes. You would have to remove the temporary sqlite-shm and sqlite-wal files as well or include them in the backup.

The delete key would also remove download items, so there might have been a problem with that key.

Interesting. I made a backup of the places.sqlite file pretty much straight away, but am at a loss as to how to retrieve the data. I didn't backup any other files though, and I just noticed that there is a file called downloads.sqlite which I didn't backup. Using an SQLite viewer, the downloads.sqlite file only has two records in (the two downloads after the failure). But if there is any way to retrieve the data from the places.sqlite file, I'd love to know.

What are the SHM and WAL files for? I didn't back those up at the time.

I don't think the delete key had anything to do with this. The only other thing I can think of is that the browser recovered a session after a bad shutdown (the computer froze and had to be restarted)... but all the entries in the list were still there when I restarted the browser and started downloading the files; it was only a moment later that they all vanished.

downloads.sqlite was used in older Firefox versions, but is no longer used in current releases as you can likely see by the time stamp of the file. So you can delete this file.