its not really the Spectre that is a black lantern, its Crispus Allen's body that trapped the Spectre's power.
and so far we know two conditions seem to exist. the way the person died determines if they become Lanterns. So yes, unfinished buisness comes into play, which is why Dove didnt come back, he was "at peace" only people that did not die in peace are coming back. the second condition is yes, the body is theirs, thats how it gets their memories.