Youtube no longer uses Flash to serve videos on most browsers, rather HTML5's Video feature. Older versions of Firefox however are an exception. You can force Youtube to play their videos via html5 player by adding this to the tail end of your youtube URL:
To have FireFox update it's Flash player automatically, you'll need to edit your existing Flash update settings.
Here is a snippet from an article by Kyle Beckman over on 4sysops.com:
When installing Adobe Flash using the exe installer, you will be prompted to select your update preference. If you use the “Allow Adobe to install updates” option, the Flash updates will install automatically.
If you go to the Task Scheduler, you can see that there is a new Adobe Flash Player Updater task that is scheduled to run daily. The task runs with System privileges. That means that even users without Administrator rights can have Flash updates installed silently without receiving prompts or requiring IT assistance.
If you selected “Notify me to install updates” for “Never check for updates,” you can go to the Control Panel and access the Flash Player Settings Manager. Click the Change Update Settings button, approve the User Account Control (UAC) prompt, and change the setting to “Allow Adobe to install updates.”
Hope That Helps!