Since you have used JW language so heavily, I will try to use JW logic while replying.
My 2 cents:
JWs think that the evil in this world is caused due to 2 main reasons:
1. Satan
2. Imperfection
During the 1000 year rule, Satan would be in the abyss, while people would be gradually moving towards being perfect. Since the source of evil is not there, what are the chances that evil would occur. There are, because humans would still be free moral agents. But very less chances, in fact negligible. However, the greatest test would occur when Satan is let out of the abyss. This is because JW's quote the following:
pe. page 182. Then, after their resurrection, it was easy for them to serve Jehovah because of living in Paradise without any opposition from the Devil. But will these billions of humans, who are then perfect, serve Jehovah if Satan is given the opportunity to try to stop them from further doing so? .....The Bible shows that Satan is successful in turning some persons away from serving Jehovah.
So, will JW's need to worry about occurrence of evil during the 1000 year reign without Satan, the source of evil, around? It would be like me asking someone, Do you now live with the fear the you or your family would get infected with small pox or spanish flu, now that those viruses have been contained?