1) You can try to see if it's working now without a restore, but protocol would be to restore it and such. I've always restored it when purchasing IMEI unlocks for my colleagues.
2) That would depend. If you've manually restored to iOS 5.1.1 without OTA and you have the 5.1.1 blobs, you can re-restore. Meaning if the iPhone 4S came with 5.1.1 from the factory right out of the box, and you never restored it back to 5.1.1, or you upgraded the iOS using the Settings.app method, you cannot re-restore to 5.1.1.
3) No jailbreak, some users report even worse battery life.