You could try & start the computer from the Snow Leopard 10.6 install-restore DVD.
From there, you could use Disk Utility from the Installer menu bar, to attempt a repair
of the hard disk drive, or see if there is a problem with the system. Hold C key down
and continue until you finally get the installer. Choose from menu bar options to use
the Disk Utility; try to repair hard disk drive. Then choose to restart from Mac HDD
by use of menu option to select startup disc.
Or perhaps even install a new system while keeping your old user accounts & files.
Even a simple start up in SafeBoot (shift key held down on startup tone, & wait for
evidence of desktop Finder to appear) may help resolve simple interface issues.
However that won't fix hard drive data errors, system damage, or hardware problems.
If you have a backup of your content elsewhere, off the computer, you could try a new
installation on the computer (if hard drive is OK) however if the computer still has an
original hard drive and has several years run-time accumulated, it may be going bad.
Or if you happened to still have the Combo update 10.6.8 file that took the base 10.6.x
to the final version (if you are running what your posted info says) that may be able to
fix the problem you see in the system preference file. And also repair disk permissions.
There likely is a more eloquent method of restoring the appearance of the Network
pane in the System Preferences section of OS X. Could be that it is not missing...
But if there is file damage or sector damage to the hard disk drive, or if the HDD
has been run near full, the file may be gone or overwritten.
To post an image in a discussion, you'd need to save a screenshot to desktop and
then direct the ASC window to get it from there. Or provide a link to another source
online that others can see in their computer, if yours is not able to upload images.
Not sure what is the cause of the problem, but it implies damage files, or errors.
Good luck! 😐