Yes, they affect the trainers post-game. For instance, if I go to battle Cynthia her Pokemon will be lv80+ instead of lv76+. This works without the need of outside Black/White 2 cartridges.
To play Challenge Mode from the start of the game you will have to have a second Black 2 and DS system or a friend with Black 2 send you the Challenge Key.
If you delete your own Black 2 and restart, you will lose the key and not be able to play Challenge Mode.