Hi Tony. The piece could have something to do with your problem, although it most likely do not. Just set your date and time to the current (disconnecting the battery resolve in having date and time set to 1. january 1970, 1:00). After resetting the date and time, reboot the phone, and your problem should be fixed. This is a iOS6 bug. Never the less, it is never a good thing to have parts left over. Please locate the part in the iPhone 4s tear-down, and return it to its proper place :-) Jon.
Of course. Just resolder it, which shouldn't be the hardest task. Regarding the blue coil; I have had success with simply just shorting the two "pads" (where the coil used to be soldered to), with some solder. Also, after the battery has been disconnected, you will have to set time and date in the phone, and reboot it. If this is not done, it will be stuck on "searching", and wont even realize if the SIM is in the phone or not. (It's and iOS6 bug). This is probably what caused your troubles in the first place. Jon.
In what phase of the restore process does the error occur? Can you read the IMEI number? Sometimes water-damaged chips loose IMEI information, and then the phone is practically useless. Btw. try change the battery. Sometimes the electronics in the battery is messed up because of corrosion. This would explain why it only starts up when it is connected. It might also explain the error you get: when the phone reboots doing restore, it will have to have a little bit of power available from the battery, because it, for just a short moment, can not drag power from the USB. Hope this helps. Jon