Problems, trying to use backup from 5.1.1 iOS Gen 3 iPod does not sync music to 4th gen iOS 6 iPod.

Question

My 3rd Gen iPod's headphone connector is basically shot, I've had it for around if not over two years now; sometimes I'm surprised it lasted this long; since I am not one who usually listens to her music without headphones, this is problematic, .
Luckily, a family memberI know had a gen 4 iPod, 32g, same space amount as my old one, lying around that they don't use anymore, thank goodness.
Well, in one case, not so much. First of all, said iPod was locked with a passcode (the passcode was guessed so many times because no one could remember it, it locked permenantly) so I had to completely wipe it. However, it wasn't so bad, though all the information was lost, I could finally use iOS 6 on an iPod.
Well, iOS turned out to be a huge problem, in the way of synicing my old backup. I have about, um, an estimated 3000 songs on my 3rd gen iPod, that of which are not all on one computer, and are from multiple computers (my desktop / laptop); I thought all the songs would transfer over with the rest of my things. I didn't think the iOS made a difference.
Lo and behold, I was dead wrong.
When my iPod was restored to my previous backup, it had no music, only my pictures and apps were added. I'd call 'it annoyed me' an understatement. I waited all that time for the ipod to restore and the backup to finish, only to get my one hundered pictures and two apps.
I have tried to look up how to extract music from my iPod, but to no avail, my computer recognizes it as a camera and I can only access the picture files; the way I read you have to do it online, you must be able to actually get into the iPod's folders, besides the picture folders, and I can't seem to do that; there was an option in earlier versions of iTunes that allowed the iPod device to be displayed as something other than a camera but it no longer seems to be an option in the new iTunes.
All the programs for extracting files, specialized for iPod file extraction, actually, cost money, which at the moment, I don't have much of, this iPod was given to me by a family member, and I'm lucky I even have it. Manual extraction is also something I've tried but failed miserably at, as mentioned earlier.
I'm really not up for re-synching one by one, I want this to be as quick and painless as possible. I have the available GB for my 3000 songs (I checked). Most of my songs are mp3s, and some are m4as. Not everything is bought, and not everything is going to be able to be downloaded through iCloud on account of the fact that there are, uh, over 2GB worth of songs, so does anyone have any solutions to my problem here? I have over 50gb in my desktop library alone, loads more in my laptop library, and I really don't feel like separating all the songs from my previous iPod into a single playlist, if there's a more efficient, free solution out there....
This would not have been a problem in the first place, if the passcode was not there; but blame the previous owner of the 4th gen iPod for that. She typed it so many times it locked her out. And so, I had to reset, but not without an automatic update for iOS 6, which my gen3 can't update to, because it's, well, outdated.

Answer 1

Connect the iPod to your computer and update to iOS 6.1.3. You need iTunes 10.7 or later on the computer.
iOS 4: Updating your device to iOS 5 or later