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I've purchased a 2009 iMac, bought Mountain Lion from the App Store and performed a Clean Install, but when I open up the App Store, I still see messages to update apps that the previous owner had installed (I don't own them), furthermore, I'm prompted to sign in with his Apple ID in order to download these updates.
I thought that when I did a clean install, all of his information should have been deleted. Am I wrong? How could I get rid of this problem?
To answer the question in your update "Can Time Machine do selective restoring?", Migration Assistant (in /Applications/Utilities) can restore data from a Time Machine backup, with some control over what is restored. See Apple's HT4889 for some details.
You also asked: "If I restore my entire home folder, will I also be restoring the faulty settings related to the crashing apps?", sadly, yes this is a possibility.
"Can I recover selective apps from Time Machine?" I don't think so.
When new updates are available on the Mac App Store, it doesn’t notify you at first. It waits “for a while” and then downloads them in the background. This includes ALL updates. System updates, app updates, all of them. When the download is complete and ready to be installed, that’s when it notifies you.
Best Answer
Now that you are on Mountain Lion, you can boot to recovery mode and re-erase the entire Macintosh HD (or whatever name the OS volume has). Perhaps your definition of clean install didn't actually clean the offending apps or you are connecting another hard drives with apps purchases from the previous owner's account. (The app store will offer to update any apps it sees, not just the ones in /Applications)
Here is a long article on recovery, but details at the bottom show explicit steps you would take to prepare an external drive to install Mountain Lion. You will want to perform those steps from Disk Utility when booted in recovery mode.
Note, that if the previous owner had "redeemed" the iLife apps (as described in the article above), those will remain tied to their account so for you to install them, you will need to purchase iPhoto, iMovie and GarageBand since they are not part of the Mountain Lion software upgrade you purchased, but after a clean wipe, you won't have those applications and could only see them if you had a copy somewhere other than the wiped drive or signed in to see past purchases for that previous account.