After system restore, no Apple applications will load

Question

I have a Mac Mini (on Snow Leopard) that I recently needeed to restore using Time Machine. I wiped the hard drive and restored everything from a good backup. The machine boots up fine (10.6.8), but when I try to run any of the Apple applications (System Update, App Store, Mail, iTunes, iWork and iLife apps, etc.), the icon bounces for a bit, and then nothing. No application starts. Other apps do seem to work (Chrome, Firefox, Boxee,.. )
I've run the Other Installations from the Snow Leopard disk to reinstall the Applications, but still the same results. I reran the iLife installation and still nothing.
What do I need to do to get the Apple applications to work?

Answer 1

Seems to me your TimeMachine drive is messed up somehow. Disconnect it
Copy your users file folders (Music, Docs etc, not Library) to a regualr external drive, export any bookmarks, product license keys or any data (program names) you require to rebuild the machine from scratch with. Write down your User name, case sensitive. Disconnect.
Hold c boot off the 10.6 installer disk, use Disk Utilities under the Installer menu to Erase > Security Option > Zero All Data the ENTIRE boot drive, it's on the far left with the drive makers name and size.
Let this complete, it's going to take some time to finish and really scrubs the drive, makes sure all the bits are working.
Next install 10.6 and reboot, setup with the same name as before and use Sofware Update to get the machine on 10.6.8
Next, install your programs from original sources, (like from the 10.6 disk) as many as you possibly can, they will run faster if they are installed before your huge files. This is also you get a nice clean copy free of issues.
Last, hook up your external drive (not TimeMachine) and return the contents of Music to Musc, Doc's to Doc's and so on into the same named user account folders.
If you keep your boot drive below 50% filled, it will operate very fast forever, just don't move it while it's running. The Zero Erase hopefully fixed any bad or damaged sectors on the boot drive that might be the reason for your problem, or it might not have been, I don't know.
Once you have a yourself a nice and clean fresh install, take this moment to erase the external drive (no data) and use Carbon Copy Cloner (free) to clone your internal boot drive to the external drive.
You can hold the option key and boot from this drive at anytime, setup CCC to auto-update the clone as a backup.
You can access the files on the clone drive too, just like any other drive.
Far as TimeMachine, you can still use that, do the same Zero Erase on it and set it up again.
Reducing bad sectors effect on hard drives
What I suspect happened is something corrupted your data on the TM drive and thus when you restored it was corrupted. Bad sectors on hard drives can do this.
Good Luck