Where do these files go?

If you put all those files and folders in the music folder, then option start and select the iTunes Library.itl file in that folder it should show all you music even if they end up having broken links. If it is blank it is because that file is not the correct library file. iTunes does not work by directly looking at your media files for the sake of building a library, it works by relying upon that file having the correct media list and file location information. For that matter you could have a library file taken from my computer and put it on yours and it would look just like my library except, of course, all the links would be broken.

By default, they’re in ~/Music/iTunes/, but can be put anywhere. If you choose a custom location, launch iTunes with the Option key held down and choose that library.

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iTunes basic functionality is: The .itl file contains a list of all the tracks you have. As you add them it notes their location and stores this. After that it does not check on the file unless you try playing it. The default behavior is for iTunes to put a copy of any file added into its media folder in a hierarchical structure. So it is possible under various scenarios (usually user-caused) for a file to be moved and iTunes' reference for it to be wrong. If everything is stored in a "portable" single iTunes folder then references are internal, but once you start splitting the library into media in one place and library files in another all the pathways to the files have to be preserved or the main library file loses track of files.

Now there may be an explanation for this but so far we have been working with several assumptions on how you stored and elected - or not - to let iTunes manage your media. I, for example, have this disabled which gives me more flexibility in managing my files and folders but in turn means I need to be on top of that management because iTunes is not doing it for me, and certainly nobody sitting half a world away is going to have any idea how it should look on my computer. So maybe there is a reason for simple solutions not working for you but we haven't really learned that much about how you set things. Normally restoring from a Time Machine backup should be as simple as dragging the old iTunes folder (the entire thing as a single item, not just the media folder, and not scattering bits and pieces over the desktop) onto the main drive into the Music folder and starting iTunes again.

If you have been managing media yourself it can introduce all kinds of complexities and then we need to re-approach the situation with that in mind.

Yes, you can rebuild a library from scratch from the media files alone. Just start iTunes. It should create a new iTunes folder in Music. Drag all the media files to the Automatically Add To iTunes folder. About the ‘Automatically Add to iTunes‘ folder. [Archived] - http://support.apple.com/HT201970 - Files put into this folder are actually moved from this folder to the correct location in the iTunes Media folder.

A library isn't selected in preferences, it is selected when you start iTunes while holding down the option key. iTunes will theen use that location to look for hte .itl library file and all the other related folders you mentioned.

Say "yes" and find the file in the subfolders in the Music folders that you wish to link to that particular track descriptor. When finished, iTunes should ask you if you want it to find others. Say yes and hope it works. If not then your media structure is really messed up. I can think of half a dozen ways this can happen, particularly if at some stage you elected to not let iTunes manage organization of your media files. If your media structure is messed up then you can either one by one re-link all your files, or you can elect to rebuild your library file and in dong so risk losing some metadata such as playlists, date addded, playcounts, etc.

Image of folder structure and explanation of different iTunes versions (turingtest2 post) - https://discussions.apple.com/docs/DOC-7392 and making an iTunes library portable.

Where are my iTunes files located? - http://support.apple.com/kb/ht1391 --> Locate and organize your iTunes media files - https://support.apple.com/HT204668