Same problem here with my 2nd Gen iPod touch. I tried all the solutions given in various places on the Internet (multiple times), but none worked. All ended in the 13019 error and no music being transferred. Even tried gettng rid of songs in the playlists that were no longer available to open because they were up to their DRM limit. Restored the iPod several times. All software up to date.
The only thing that worked was going into the iPod settings and erasing the entire iPod and setting it up as a new one. I skipped trying to erase and then update from a previous backup just in case the backup was the problem. Now music transferred and it works again.
I had never had this problem before, and I'm thinking it might have to do with the fact that it was really dead. I hadn't used it or charged it since I got my iPhone 4S back in October. So I don't know if being that dead had anything to do with why the error occurred or not.
But it syncs music now, even though I'll have to add all the apps and other things back in with subsequent syncs.