I think Ive figured out why the entity kills? the resurrected heroes/villeins (well the recent ones anyway) once they've completed their tasks. 
Is it because they are now at peace with their lives so they can finally rest in peace. Its just a thought since Dove couldnt be resurrected or become a black lantern because he was "at peace" according to the white lantern and the black rings. 
If the true reason has been reveled then please don't be too hard on me because its been a few issues since ive checked up on it :p

Am I the only one that hopes Luthor will get the white ring?

but dc just said today or yesterday that Geoff johns was going to write a ongoing series about aqua-man fowling the events of brightest day

and hawkman and hawkgirl completed their tasks and they didnt die