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You can try to fish the wire out with a hook or forceps but usually the insulation makes this extremely difficult. If you pull the appliance out you can remove the rear access panel to get to the wire and get it back into place.
You can't use it with the wire off, no.
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There should be an access panel on the back as that is where the convection motor is.
Just for the wiring, yes. Usually the wires do not pull through, so you would only need to access the back if they do. It must be your "lucky day." :)
If you put the old element back in. You don't want that wire shorting to anything.