While itunes 12.3 won't work with the original ipod touch (3.1.3), at least on Windows 7, itunes 12.1.3 works fine, syncs and backs up.
I had to read this thread to find out why itunes wasn't backing up my old ipod. Since it did back up with itunes 12.1.3 I went back to that version.
However (again, for Windows 7) if you're going to go from 12.3 to a previous version of itunes you have to follow this guy's advice:
https://www.youtube.com/watch?v=Sn82w_hxgwQ
and uninstall itunes and then uninstall the other four Apple programs that itunes leaves behind before you install 12.1.3, otherwise 12.1.3 will still not back up your ipod (and they may keep itunes from installing if you're using a version older than 12.1.3.)
Also you have to follow everyone's advice about having to replace your itunes library file with a previous itunes library file that's old enough to be compatible with the version of itunes you're installing.
That's the advantage of using 12.1.3 instead of say 10.7, you can use a more recent old library file.