Henry...
Something is apparently wrong with your Windows installation, since you are unable to install Flash Player.
Is it fully updated with all patches from Microsoft Update? How about your Firefox version? Is it up to date?
You may want to check for malware to see if some kind of "nasty" is causing your problems. Adware is one thing I'd suspect.
First of all, run adwcleaner to make sure adware isn't causing your problems. It's great for removing most common adware that can cause issues with browsers and extensions (like your inability to get the flash player plugin installed for use with Firefox). Use it's scan choice, then mark everything found and use the clean button to remove the adware found during the scan.
http://www.bleepingcomputer.com/download/adwcleaner/
Then, run Malwarebytes Free. It's a super product that can find and remove most common malware and adware that many other products will miss.
https://www.malwarebytes.org/antimalware/
Make sure to mark everything it finds (including PUPs, which are Potentially Unwanted Programs like adware left overs that could be causing issues with your browsers) for removal, as they're not always flagged for removal by default and you have to check the boxes beside them in the scan results.
Then, I'd run a followup scan using Hitman Pro. It's a great scanner using multiple scan engines, and you can use it for free and remove any malware (and awdware) found for up to 30 days after you install it, without anything running in the background that could interfere with the antivirus products you're already using. Same deal... remove everything it identifies. Get it here:
http://www.surfright.nl/en/hitmanpro
Many antivirus programs will not flag adware as malicious by default, because adware is usually something you give permission to install. Most downloads of free software from popular download sites like cnet's download.com have adware included by default now, unless you're very careful to use a custom installation and uncheck the boxes to include it. They count on users not reading the "fine print" that indicates adware is included by default unless you take steps to "opt out" by unchecking the boxes for it (and the installers can be confusing and misleading, too).
That's just one of the trends we're seeing lately so that download sites and software developers make more money on their "free" products. A lot of it can cause issues with browsers. So, make sure to "opt out" of any extras (especially browser search toolbars and similar), when installing software to avoid adware infection.
Then, after you've cleaned your computer of any malware and adware that may be preventing flash player installation, try to reinstall flash player again and see if it works with firefox on youtube.
As for google chrome, it looks like youtube starting forcing the use of it's html5 player in May (although Firefox should still use adobe flash player by default unless you've "opted in" to using html5 only via a setup page google has for that purpose). More about that kind of thing here:
http://googlesystem.blogspot.com/2014/05/youtube-switches-to-html5-player-in.html
IOW, by default, if you have flash player installed in Firefox (which you don't because of something wrong with your Win 8 installation preventing you from installing it), then it will be used for youtube playback unless you specifically change the settings on the setup page mentioned by that blog post. But, with Chrome, html5 is now the default player.
There is an extension that allows you to force the use of Flash Player instead (already built into Chrome), at least for now. This one:
https://chrome.google.com/webstore/detail/disable-youtube-html5-pla/enmofgaijnbjpblfljopnpdogpldapoc
But, I'd clean your PC of any malware and adware using the above tools (adwcleaner, malwarebytes free, hitman pro) first, as the html5 player may work OK in chrome after that cleanup process gets rid of any malware or adware causing issues.
Then, you should be able to install and use flash player for youtube playback in Firefox after correcting problems with your Win 8 installation (and I'd expect adware or malware to be a potential cause of your issues that the 3 free tools I linked to should be able to find and eliminate for you).
If that still doesn't solve your issues (cleanup of any adware and/or malware that your existing antivirus product didn't detect and remove), then you may need to do a repair installation of Windows 8 to get a working system again unless you have a good disk image backup of your installation from before the problem started that you could restore. Here's a page outlining some of the steps you can take if it comes to that:
http://windows.microsoft.com/en-us/windows-8/restore-refresh-reset-pc
But, I'd try the steps I mentioned above first for starters, as I suspect you've got some kind of adware or malware causing your issues that the products I linked to can probably find and remove for you.
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JimC
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