If an enemy occupies a square can it still be successfully searched or can an objective be retrieved (from under the enemies nose).

The rules do not prohibit it , however I would have thought either side may want to place a "guard" in such areas as a valid strategy. 

Unless the enemy is carrying said item, or the quest rules state otherwise, the item is fair game.

I agree with Rico.  Searching can  be performed from an adjacent space, so that part is definitely valid.  Doing things with quest objectives is usually outlined int he particular quest, so it may vary, but unless the quest rules say that you must be on top of the objective marker (not adjacent), then you're golden.

This doesn't mean that it's pointless to post guards, mind you.  Even if a hero moves in, searches and moves out again (interrupting his move action to search or perhaps using fatigue to move), he probably won't get too far away after searching, so whatever guard monster was there can follow him and attack.  I'm pretty sure the OL also has a couple of cards that let him spontaneously attack with a monster when a hero moves too close (I don't have the OL deck memorized or anything, but that seems familiar to me.)

Interesting… We were playing a lot with rule that you cant search under enemy figure… We will search rules for this…