I spent an afternoon trying to troubleshoot Itunes Error 4013 over the weekend.
Of course I tried the typical troubleshooting steps shown in all the videos on the internet.
Nope.. none of them worked.
Final guess... USB port is toast.. or Proximity Sensor might be dead.
NOTE.. during the itunes restore/update... it never gets to the part where it should transfer over the firmware file. This means.. it can't send over data for some reason????