I believe I've nailed down the problem, and it's only peripherally related to the metadata.
The Kindle Fire HDX doesn't like PDF files that have an "&" ampersand character in the filename. 
If there is an ampersand in the filename, like when there are two authors, the file is shown in the list, but it will not open.
If the file has the NEW icon, it goes away, so it's clearly looking at the file and then returning to the list without showing it.
If you edit the metadata in Calibre such that the ampersand goes away in the filename when you send to the device again, then the file will open properly.
This doesn't appear to be an issue for MOBI or AZW3 files, only PDF.
So, not really an issue with Calibre so much as with the Kindle's software, although it might be a good idea to have Calibre do a character substitution in Calibre the way it already does for colons and certain other characters.
This doesn't address the question of why I ended up with PDF files that wouldn't open in Acrobat, but I saw that early in the whole process of researching this, and so far I've not been able to reproduce it. Not gonna worry about it until I do.
