Hmm, that is indeed interesting… As far as I know, Mac OS X v. 10.4 should be able to "see" any file just fine without requiring to boot from the drive. Indeed, even if a drive needs to be booted from for its contents to be used (if that drive contains some system files that you want the computer to load, for example), a file still is a file and should at least be displayed — especially files as technically mundane as your Documents or digital pictures.
When launched, applications will read files and folders they have placed in your Library folder, within your home. These files are usually located in the Preferences and Application Folder support and store your settings as well as, in some cases, database information. When booted from your computer's internal drive, your home is the one on this very drive, not on the external one and, as such, your applications will not go and look on your external drive.
To restore these applications to their original state, quit them, copy the files over from your external drive to the internal one and re-launch them — just be careful to not copy damaged or corrupted files and do not copy over the contents of the "Caches" folder.
As a general rule, avoid copying the applications themselves from the external drive but download a fresh copy. Some applications will be better off if you re-configure them manually as well, although you are the one who will decide between keeping it clean (the manual way) or practical (copying files over).
About files missing, this is definitely strange, provided the cloning operation went well. Before clean-installing Tiger, did you ensure the files were all there? Could you have deleted them from this drive by accident?
If not, they may be hidden in a former FileVault (simply open it to find them) or tucked away somewhere unexpected (in which case Spotlight should help you).
You may want to use Disk Utility to verify and repair your external drive before attempting any copying operations.