I just need ONE thing. (Mac 10.5.8)

What version of OS X are you running? If it's not 10.5.8 there's a chance that the pref pane won't run at all. And if it does run, there's a chance it could muck up something.

 

Corners?
There's nothing called Corners in the Appearance pref pane.
Do you mean Hot Corners - which is in the Desktop & Screensaver pref?
Does that pref pane work, or are you having real problems with other preference panes, too? If so, using a pref pane that you get from someone else, even if working, might still NOT work when you try it in your system.
(because the problem is not the individual pane, but your system

BTW, you also should not find that Appearance pref pane in the location that you asked about ( the PreferencesPane folder doesn't exist) - it IS found in /System/Library/PreferencePanes folder, if you need to look in there to check.

 

Your best way to try to get the preference panes back with minimal disruption is to download and install the 10.5.8 Combo Update. And the ones you are missing would be in /System/Library/PreferencePanes/ - the folder you listed is for the user installed pref panes.

 

Your system preference panes are NOT in that folder from your screen shot.
You are looking in /Library - please check in /System/Library/PreferencePanes

But - go to your user Library (in your home folder), then open the Preferences folder, and delete the file "com.apple.systempreferences.plist", and "com.apple.preference.desktopscreeneffect.plist"
Then, log out and log back in, or simply restart your Mac.
Even if your system preferences still do not appear properly, the hot corners will be reset to defaults - none - which will fix the corner issue that you are struggling with.

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I suspect that the pref pane (if it really is missing) may not reinstall just by running the combined updater, although it may work.
BTW, the pref pane file that you will be looking for is named "DesktopScreenEffectsPref.prefPane". That shows up in the actual Preferences as Desktop & Screensaver, because that pane supports several different functions.

 

I did say "try". The next thing to do if that didn't work would be reinstalling 10.5 and updating from there.

 

Agreed - the combined updater can be a great fixer, and I probably mention it at least twice a week (for the last 10 years )
Hopefully, the OP will look in the correct folder, and find the pane still there. Clearing settings/caches will likely get those to reappear, in that case.