Clear the Cache and remove the Cookies from websites that cause problems via the "3-bar" Firefox menu button (Options/Preferences).

"Clear the Cache":

• Firefox/Tools > Options > Advanced > Network > Cached Web Content: "Clear Now"

"Remove the Cookies" from websites that cause problems.

• Firefox/Tools > Options > Privacy > "Use custom settings for history" > Cookies: "Show Cookies"

You can reload web page(s) and bypass the cache to refresh possibly outdated or corrupted files.

• Hold down the Shift key and left-click the Reload button
• Press "Ctrl + F5" or press "Ctrl + Shift + R" (Windows,Linux)
• Press "Command + Shift + R" (Mac)

You can remove all data stored in Firefox from a specific domain via "Forget About This Site" in the right-click context menu of an history entry ("History > Show All History" or "View > Sidebar > History") or via the about:permissions page.

Using "Forget About This Site" will remove all data stored in Firefox from that domain like bookmarks and history and cookies and passwords and cache and exceptions, so be cautious. If you have a password or other data from that domain that you do not want to lose then make sure to backup this data or make a note.

You can't recover from this 'forget' unless you have a backup of involved files.

If you revisit a 'forgotten' website then data from that website will be saved once again.

Sorry to take so long to get back to you. First, in my original question I forgot to mention that I am using Windows 10. Regardless, as I mentioned in my query I had tried everything you suggested before posting this question (except for "forgetting" a site) -- but to be fair I tried every single step you suggested again, several time each, with no satisfactory results. I'm still getting the "Your connection is not secure" (in other words SEC_ERROR_UNKNOWN_ISSUER) on many domains/sites and it shows up inconsistently. If I leave the default home page to be Mozilla search I don't get the alternating error (with Google and other potential home pages which are https I get the error every other time I try to access the page). Also as noted, every mozilla.com subdomain other than www.mozilla.com shows up as if JavaScript has been disabled i.e. all text and none of the links do anything useful. Other than not being able to get help/support from within Firefox (I'm using MSIE to do these posts) or to get extensions or add-ons -- YouTube is also being treated as if JavaScript is not enabled (it is, I checked in about:config) and this is the one site that I really use Firefox a lot with (I tried having Firefox "forget" YouTube but it didn't help). Plus I use FireFTP and Firebug for my work and can't get the extensions now. >>> One clue that there is a weird problem in my FF installation -- in about:config under permissions the permission "dom.apps.reset-permissions" is set to "user set boolean true" -- if I right click and hit "Toggle" it gets set to "default boolean false" and stays that way until I close Firefox and then restart FF which seems to reset it back to "true" (every time I set it to "defaut false" then restart). I use Esset Security Suite -- could that be causing that phenomenon -- possibly even be the source of all my Firefox problems? I hope some of the new info I have provided may give a clue as to what's going on. I have always recommended that my students use Firefox as their default browser because of its versatility -- but now, not entirely sure that I can continue to be comfortable or confident doing that.

Did you check the issuer of the certificate to see if that is issued by your ESET security software?

ESS 9: Advanced Setup -- Web and Email --- SSL/TLS

Root certificate then uncheck "Add the root certificate to know browsers"
SSL protocol: Do not scan SSL protocol

Thanks very much for a workable solution. I don't really want to remain exposed without SSL scanning if possible but I tried turning it back on and YouTube is screwed up again. How dangerous is it to continue browsing with it disabled?

Thanks again.