Just to make sure there is no confusion having a song file is different than having it listed in your library. iTunes doesn't automatically delete a file from the library if you move or manually delete the music file. Also adding a music file to your computer means you still need to go to iTunes and use the Add option under File to add the song(s) to your library. There can be exceptions to this (like when you have iTunes "rip" [aka import] a CD) but remembering this should avoid a lot of confusion.
Arrange the files as you want on the computers. If they were in iTunes before and you have moved them then you either need to delete them from the iTunes Library and add them again OR go to the info or properties for each song and tell it where the file is now. Unless you have things like ratings, etc you want to keep in the Library the first option is easier and quicker, especially if you have a number of songs.
Duplicating or having a copy of the music files is a good thing but I would suggest it is best and wisest to not duplicate files in the Library. When you move the music files it is a good idea to do one of the steps above.
You can make sure a song is in your Library by looking for it. They are easy to add using the option under the File menu to either add a file or add a folder. When you look in the Library you can see it is there. If you double click on it or highlight and press Play then that is an easy way to make sure it has valid info. If there are duplicates then trying to play each can be a way to find out which is good, after you have the music files moved and set up as you want. You can use the appropriate steps above if you find a dupe, song without a file, or are missing some songs in your Library.
Let me know if I overlooked one of your questions or if you have a question about any of this.
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