screenshots of some affected sites (YouTube and NFL.com)

If you use extensions (Firefox/Tools > Add-ons > Extensions) that can block content (e.g. Adblock Plus, NoScript, Flash Block, Ghostery) then make sure that such extensions aren't blocking content.

You can try these steps in case of issues with web pages:

Reload web page(s) and bypass the cache to refresh possibly outdated or corrupted files.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

Clear the cache and cookies only from websites that cause problems.

"Clear the Cache":

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

"Remove Cookies" from sites causing problems:

• Firefox/Tools > Options > Privacy > Cookies: "Show Cookies"

Start Firefox in Safe Mode to check if one of the extensions (Firefox/Tools > Add-ons > Extensions) or if hardware acceleration is causing the problem (switch to the DEFAULT theme: Firefox/Tools > Add-ons > Appearance).

• Do NOT click the Reset button on the Safe Mode start window.

• https://support.mozilla.org/kb/Safe+Mode
• https://support.mozilla.org/kb/Troubleshooting+extensions+and+themes

Everything is pretty much back to normal now (Shift-reload did the trick), but videos on some sites (YouTube, wftv.com) still won't show up. These videos use the Adobe Flash plugin, which I reinstalled, to no effect.

WFTV videos have come back on their own, and I got YouTube to work again by adding 'https://" before its URL, but embedded YouTube videos don't show (on http://failblog.cheezburger.com/fails for instance) without it included in the embedding code.

Check that - embedded videos are back, too - problem seems to be solved.