1) 4A quests requires to put a wight out of the encounter deck into the staging arena. It goes into the second staging area right?
2) What do I do with this wight if I beat 4B without engaging it?
3) How do you manage that massive ally discard if Great Barrow appears from the encounter deck more than one time?
4) Use Frodo to shuffle Great Barrow back?
1) That is correct. The When Revealed effect on stage 4A is what causes you to add a Wight enemy into the staging area. Since Stage 4A is what creates this effect, the enemy goes to Stage 4's staging area.
2) If you beat the stage without engaging the Wight enemy, the enemy will be combined with the other staging area.
Combining Staging Areas
When a player is instructed to combine staging areas with another staging area, each enemy and location card in that player's staging area is added to the other staging area. Enemies engaged with a player remain engaged with him.
3)It is quite difficult to manage it multiple times, but it helps to have effects that let you put allies into play from your hand (like Sneak Attack). It also helps to save resources on your heroes so you can repopulate your army (though be careful of that one treachery that makes you raise your threat by the number of resources you have!) This quest is generally considered to be difficult.
4) I believe the Great Barrow's immunity to player card effects will prevent you from using Frodo's ability to shuffle it back into the encounter deck. I do not have a good source for this, though.