Your "More system details..." shows Flash Video Downloader Youtube Downloader version 3.6.8. I noticed there is an update available on the add-ons site today. Maybe that will help?

Edit: The description of the update is "Fixed Youtube Download". https://addons.mozilla.org/en-US/firefox/addon/flash-video-downloader-youtube/#detail-relnotes (click to expand the version information)

hello, users of different downloading addons have reported similar issues with youtube recently - it's likely that youtube has implemented changes on their server-side that causes those extensions to break. please visit the website of the addon to inquire if there's already an updated version available or otherwise directly contact the developer of the addon and report the issue.

I've already run through the usual reset, update, de-activate, re-activate, turn off everything else routine and nothing made any difference. However just after posting my question I was able to download a video from YouTube by making sure that right before I clicked on the Flash Video Download button in the menubar I cleared all cookies. The video then downloaded normally. If I make the mistake of clicking the DL button with even one existing cookie then there is no way to fix the problem, it is a continuous loop of the dialog box and no way to fix without a restart of Firefox. This is certainly a new development and no doubt due to something YouTube has changed on their side.

I don't allow many cookies and delete them every few minutes and always automatically delete them when Firefox closes, but this behavior only started within the last 48 hours and as I mentioned seems to be independent of my machine, OS, and browser version. It is limited to YouTube only though. I just wanted to post this because I'm probably not the first to encounter this problem and I'm sure others will see it soon. Thanks.