DenP, the Docs app has been acting kinda odd for me too. I'll see mobi books that are definitely downloaded with no indication that they've been downloaded (no tic) even in the All category and none at all in the Downloaded section even after several syncs. And when I tap on the book, it does nothing (doesn't open and doesn't download). This has been going on since the last update of 126.96.36.199
When I go to Books and sort by Collections and open the Collection, the mobi book does show as downloaded and I can open it. Also, if I click on the small square that has prior app windows under it, I can select the one that had the mobi book open and it will open. It's very odd behavior all 'round.
Thanks for that update, CBR. I don't read books on my Fire but I have a lot of personal documents such as Instruction Manuals and spreadsheets, most of which I've sent using "Send to Kindle for PC" and so far I've not had a problem. Perhaps I have that to look forward to.