iPhone4 unlocking questions

I have a few questions before I get my 4 unlocked. It is currently running 6.3 and jailbroke via redsnow 6.1.3. I will be going through my carrier. Questions being..
1. Can it be unlocked if jailbroken?
2. After approved and I do the back up/restore via itunes, can it stay jailbroke and also can it stay at 6.1.3 or will it automatically install iOS7?

Any advice on this is welcome.

 

Well, upon receiving all the good advice, I decided to just go ahead and do it. Phoned the carrier and they sent for the unlock. Told me it would take 2-3 hours. 10 minutes later I plugged the phone into itunes with an unregistered sim and it asked for activation. Put an registered sim in from another carrier and it picked up network and worked straight away.