What if I have done all of these steps and find that the phone won't charge. As soon as I plug the phone into anything, an ipad, mac or wall outlet, it cycles to the recovery screen.
What I have: the phone is off, latest itunes installed but not running, either on the mac or windows laptop, the phone is plugged into the machine, the phone shows the battery, flashes to the apple logo then flashes to the recovery screen, at which time itunes boots up and I go through the restore cycle until all of the programs are removed and the download of the ios begins and instantly gives me the error 2001, which you addressed.
Or: the phone is off (?dark) I plug it into a charging unit, (directly into mac, transformer or win laptop). The battery displays, the apple logo displays then the recovery mode screen comes up. I believe the ios is too screwed up to let the thing take a charge in the recovery mode.