iTune Files and Folders changing locations without my say so

See Make a split library portable - Apple Community for more on organizing iTunes on an external drive. Note that changing the media folder does NOT tell iTunes where to find files that you've moved by hand. It tells iTunes where it should put new content that it is responsible for organizing, that includes new downloads, transcodes, rips, or anything consolidated from outside of the media folder into it. iTunes will always put this new content in its preferred layout not the one you choose. Given you're on a Mac once media is inside the media folder you can move/rename it and iTunes should keep track of the files. Turn off the Keep Organized option to stop iTunes moving anything when it is played or updated.

I would concur with the first steps suggested above that you put everything back where it came from to make the library work again. Once you've confirmed that the library is also in the portable shape, iTunes folder, containing iTunes Media folder, containing all media in any layout that you like, then move or copy the entire iTunes folder from source drive to the destination drive and hold down option as you launch iTunes to choose the library from its new location. The media folder setting will update automatically and the library should work without your files being moved within the media folder.

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You cannot manually move music from one hard drive to another - the iTunes application isn't 'smart' enough to find where you've moved it. Assuming you have playlists you don't want to lose, to fix this, the first step will be to copy all the music back to its original location and delete it from the external drive. (Quit iTunes first) Once the copy has finished launch iTunes and make sure it "knows" where the music is. Then in iTunes Preferences select Advanced. You'll see a box labeled iTunes Library Location. Click on the button labeled Change and navigate to your external drive where you want the music to go. Having done that now select consolidate. The music will be moved to the new location.

If you don't care about playlists and don't want to spend the time having the music copy back to the original drive, you can open the Music Folder, then iTunes folder and then delete all the files (but not folders) labeled iTunes. Now launch iTunes. As described above, open preferences and select the folder on the external drive where you want music to go. Now drag the folder containing the music onto the iTunes icon on your Dock. This will cause iTunes to reindex your music so it knows where it is on the external drive.