Federico, it does sound like battery problems. I would most certainly stat with that. If nothing else, it will give you a clean starting point for further trouble shooting. Here are the guides for your phone and the battery is available right here as well as many other places. Also make sure that you try a different cable and that you clean your dock connector of any lint and corrosion. Hope this helps, good luck.
To long for a comment :-) The Error 1600, 1601, 1602 as defined by Apple :"
Follow the steps listed above for Error 1604. This error may also be resolved by disabling, deactivating, or uninstalling third-party security, antivirus, and firewall software. See steps in this article for details on troubleshooting security software." and here is 1604 "This error is often related to USB timing. Try changing USB ports, using a different dock connector to USB cable, and other available USB troubleshooting steps (troubleshooting USB connections. If you are using a dock, bypass it and connect directly to the white Apple USB dock connector cable. If the issue persists on a known-good computer, the device may need service.
If the issue is not resolved by USB isolation troubleshooting, and another computer is not available, try these steps to resolve the issue:
Connect the device to iTunes, confirm that the device is in Recovery Mode. If it's not in Recovery Mode, put it into Recovery Mode.
Restore and wait for the error.
When prompted, click OK.
Close and reopen iTunes while the device remains connected.
The device should now be recognized in Recovery Mode again.
Try to restore again.
If the steps above do not resolve the issue, try restoring using a known-good USB cable, computer, and network connection." from here. Reason for the battery change is that the iDevices seem to habitually doing weird things when the battery is no longer holding a charge etc. I have got the same error on a few phones with bad batteries, that is why I replace these with some of the more dubious errors. Especially the way yours has been acting reminds me of this. Also, check on Try to restore get a error 29 Try what Majesty is suggesting. This does work for most of the iTunes errors.