MacOS – Purchased Logic Pro X not showing up in updates

Yes! When you open the App Store's Updates tab, you'll see a special dialog allowing you to accept the apps such as iPhoto that were bundled with your computer. If you accept them and enter your account info, the licenses will be automatically associated with your account. In the future you'll be able to receive updates exactly as if you had purchased the apps.

(The details probably depend on which computer you have and when you purchased it. But if your computer is a relatively recent model then you probably will have access to these apps. If you want more details you may need to contact Apple.)

There is no way to make App Store output debugging info, all debugging info is that what you see in the Console logs.

App Store doesn’t “detect” that it is on Mountain Lion. Mountain Lion 10.8.0 ships with a different version of App Store that is unique to that version. In addition to this, App Store may check the SystemVersion.plist file to determine on which version it is running.

iWork and Mountain Lion don’t appear in the Purchased tab of App Store, because you did not purchase them through the App Store.

As for the App Store not detecting the 10.8.1 update, that is a problem even I have had (With Xcode, though). About 3 days after the update disappeared, it reappeared. I never found out the cause of the problem.