But unofficially unlocked iPhones can't be updated or else they lose the unlock. You always have to wait until an unlock is available for a given version before updating.
Bottom line: If you don't need the unlock, don't bother with it. If you do need the unlock, look for an officially unlocked iPhone instead.
I would suggest putting a question mark with the thread title.
Yes you can save and backup as normal. What you won't be able to do is upgrade immediately when Apple releases an iOS update. You will have to wait until a jailbreak/unlock is released.