Why do apps which are not installed run from an external bootable drive?
Migration Assistant copies everything, including corrupted preferences.

If you want to have a truly clean system, what you need to do is do a secure reformatting of the hard drive that you will be installing the new system on, make sure that you only install applications directly using the installer from the original installation source (and if available, you should get it from the Mac App Store).

On Mountain Lion, the OS will recognise apps installed on an external drive in a previous system as installed on the System. You need to uninstall or delete those apps and then they will not be seen as installed.

As regards the apps you cannot launch due to the GateKeeper security in Mac OS X 10.8 Mountain Lion, that is part of the enhanced security. If you try and launch those apps from within PathFinder, you will not be able to. The first launch must happen in the Finder, and if you get an error telling you that the app cannot be launched due to security settings in GateKeeper, then you launch the app by right-clicking on the app and selecting "Open" from the contextual menu. Then you will get a new dialog box that asks you if you are certain you would like to open the app as it is not signed/recognised.

You can disable that feature by changing the settings in the Security &amp; Privacy Control Panel, and setting "Allow applications downloaded from:" to Anywhere. The default setting is "Mac App Store and identified developers"