1- Cards with victory points are only added to your victory display if you defeat or explore them, so if you draw them as a shadow effect they do not go to your victory display.  

2- I don't know what card you are referring to, but as long as you are just searching your deck it doesn't count as drawing.  Gandalf's Search says "Search the top X cards of your deck. Add one of them to your hand.  Return the rest to the top in any order." Or something like that.  So you could use that card, as it says "Add" and not "Draw".  If you are talking about Imladris Stargazer or tactics Bofur, than they are perfectly good cards to use, as you are not drawing, just looking at the top five.

3- Guarded objectives are guarded by the first card that you draw to guard them, as long as it isn't another objective.  So if you pulled a treachery to guard an objective, it would resolve, and then the card isn't guarded anymore. If you drew a location to guard an objective, you would have to explore that location in order for the objective to be unguarded.  Even if the location is the active location, it is still guarding the objective until it is fully explored.

4- Unfortunately, no.  Those nasty objectives stay with you the whole time, so you are pressured to get the quest over with quickly after you attach them to your heroes.  The only time they would unattach themselves is if the hero that they were attached to died, and then they would go back to the staging area. 

Hope this helps!