There's a reason why Apple appears to onlookers to be so anti-Flash, and it's because Flash sucks on Macs (it may also suck on other OSs, but I don't use them enough to be knowledgeable).
What to do about it depends on what browser you use (your question doesn't say, as Mozilla's Plugin Check isn't FF-only):
- Safari - the answer is ClickToFlash. Honestly, C2F is something that should just come pre-installed on every Mac.
- Firefox/Camino - As Ken mentioned in a comment, you want the FlashBlock extension.
- Chrome - use the FlashBlock for Chrome extension.
As has already been said: you can get along fine on almost all sites without Flash. Here's some additional recommended reading, both of which are from John Gruber at Daring Fireball:
If you don't have any Time Machine backups, now would be a good time to make one.
The Desktop Preference in your screen grab is displaying the contents of Macintosh HD -> Library -> Desktop Pictures which appears to be empty. If it really is empty, you may be able to locate, replace, or download the missing files and that could be a simple fix. But if there are other issues and it seems that Mac OS X has been damaged, I would think about using Recovery HD to reinstall the OS. User pictures are located in the /Library/User Pictures folder. Since Disk Utility has verified and repaired permissions on Macintosh HD, I guess those files are missing too.
Step one: make a Time Machine archive of your Mac Mini on an external drive. Once you are sure it is properly backed up, you could try entering this in Terminal to search the Library folder for the Mountain Lion default Desktop picture:
sudo find /Library -name "Galaxy.jpg"
Or search the entire disk like this (it will take quite some time):
sudo find / -name "Galaxy.jpg"
And just for good measure, run a bit of system maintenance:
sudo periodic daily weekly monthly
If you are still in the same predicament and decide to re-install Mac OS X, reboot holding down the Command and R keys until you see the Apple logo appear. You should see a list of utilities including Disk Utility (try it one more time) and "Install Mac OS X" which is a 4GB download, so a good network connection is going to help. After a clean install, you can use Migration Assistant to import Users from the Time Machine archive you made earlier. Applications are best re-installed from the App Store, original media, or a source you can trust.
Best Answer
While it's a bit dated, the below linked document may help address your question. This letter titled Thoughts on Flash was written by Steve Jobs in April 2010. While it was written to address why iOS doesn't support Adobe Flash, few points are also relevant for reliability, security and performance on macOS. I have quoted them below:
As stated by Apple, security and reliability concerns, and availability of open and standards compliant alternatives have propelled them to discourage use of Flash. Flash still continues to be available for macOS for those who need it.