iPod Touch 4g won't restore - errors 1602, 2001, 2006

Please be kind to this not-so-technologically gifted one!

I have an old iPod Touch that has been sitting in the drawer for a while. It would be useful, even if plugged in, if I could get it to run again. It was sentenced to the drawer because it would get in the reboot loop, where the apple logo would flash.

Now, if I plug it into the wall with an Apple cable/plug usb adapter, it will go to the connect to itunes screen/ Every once in awhile, it will go black for a few seconds, and then return to the connect to iTunes screen.

If I plug it into the wall with an Apple cable/samsung usb adapter, it has a black screen.

If I plug it into my MacBook Pro 2008, I get the black screen. I can put it in DFU mode, but iTunes gives me error 1602 or 2001. I've tried three different Apple original cables in all of my USB ports - all appear to be in good condition. All software is up to date.

I took it in to the Genius bar, thinking it is likely my ancient laptop that can't make the connection, hoping that they could restore with one of their laptops. The rep didn't want to try to restore it, saying that the battery isn't holding enough charge (though I don't need it to run on battery - I could use it plugged in).Is

Would buying a battery on amazon get this running? Would 8GB be enough for today's iOS and a few apps - facebook, a few games for my daughter? or don't even bother?

Thanks!