Hi MrSulfur,
Wow, this sounds like a mess. But think it will turn out to be fixable. One thing to understand, just by way of getting a picture of all this, is that iTunes is designed so that you never have to be messing around directly with files and folders on your computer. In fact, moving files around will only confuse iTunes, because the app has no way of "knowing" that something has moved from one place to another. This may be the source of the "original file could not be found" error messages.
The normal settings for iTunes -- you can check this in iTunes > Preferences > Advanced -- are "Keep iTunes Media folder organized" and "Copy files to iTunes Media folder when adding to library." If those boxes are checked, then iTunes is set to handle all the filing, naming, folder-creating and so forth. It stores all this information in iTunes Library.itl. When you add songs from the desktop, for example, you should drag them into iTunes itself (either into an open iTunes window or onto the icon in the dock) so that the app can update iTunes Library.itl, copy the music file to the iTunes Media Library folder, create a new folder or folders if necessary, and generally keep track of everything. At this point, you can safely delete the original file from the desktop or wherever it came from.
Straightening all this out should not be too hard, though it may be time-consuming. When you get an error message that says "Would you like to locate it?" you can navigate to the proper folder and select the song. The folders will be organized by [Artist] > [Album] > [songs in numerical order]. (The exception is multi-artist compilation albums, which are filed under Compilations > [Album].) If this is too tedious, there's an applescript to help automate the process by locating all the songs in your Media folder that are *not* listed in your iTunes library: http://dougscripts.com/itunes/scripts/ss.php?sp=musicfolderfilesnotadded2
The problem of duplicate or triplicate songs is a bit different. In my experience, some of these may be genuine duplicates (i.e., different files, but copies of the same song) while others may be multiple listings of the same file that, for some reason, has gotten re-imported into iTunes Library.itl. You can check this by selecting the song, then File > Get Info, and click the "File" tab. At the bottom, you'll see the name and location of the music file on your computer. Now you can click the left/right arrows to flip to the other versions of this song. If the file names and locations are different, then you've got duplicate copies, so you can delete all but one of them. If they're the same, then it's the same file with multiple entries. You can delete all but one copy from iTunes, but when you get a window asking something like "Would you like to move this file to the Trash?" select "Keep File." I hope all this makes sense!