1. The Stalker remains hidden as a card until he's spotted, casts a spell, or goes out of hiding to run. Remember to put hidden markers over your dummy cards to really fool your opponent. A dummy card with a hidden marker is revealed automatically if it comes within 12" of an enemy.
2. Your second question is trickier. If the model was hidden, then spotted, then runs/moves around the corner of the building, I would say no. The enemy knows you've gone over there. You'll have to make another move and then hide to pull that off.
If the model was out of LOS, approached the corner of the building within 6" of the enemy, but still out of LOS, then I would say it could hide.
3. Yes. Read the rules for spotting hidden models. Over 12" there is a -4 penalty to spot checks (check my memory) and terrain modifiers too.