Then Player A proceeds to cast a 1/1. Does the 1/1 die because the opponent's -2/-2 is applied first, the 1/1 dies and then the player A's +2/+2 is applied? Or do the Elesh Norn -2/-2 and +2/+2 just cancel out each other?
If Player A and an opponent both control an Elesh Norn, a creature that's normally 1/1 will enter the battlefield under Player A's control as 1/1, as the two relevant Elesh Norn effects — +2/+2 and -2/-2 — cancel out (neither effect depends on the other) (C.R. 603.6b, 613.4, 613.1g, 613.3c, 613.6, 613.7).
EDIT: Correctness edit after comment 3 was posted.
These effects are actually applied one after the other, the order depending which Cenobite entered the field first (rule 613.6), but it doesn't matter: only the net sum of all such effects is relevant (rule 613.4).
613.4. [...] All resulting changes to an object’s characteristics are instantaneous.
Then Player B plays his own Elesh Norn.
Then Player A proceeds to cast a 1/1. Does the 1/1 die because the opponent's -2/-2 is applied first, the 1/1 dies and then the player A's +2/+2 is applied? Or do the Elesh Norn -2/-2 and +2/+2 just cancel out each other?
EDIT: Correctness edit after comment 3 was posted.
RULES OF MAGIC :
http://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules