Where is the the Great Hall Treasury Chest for spoils of the assigned adventures?

Solution 1:

Your great hall is in the northeast corner of the stronghold map.

Once inside, head towards the steward and enter the doorway to the right. This is where the great hall merchant appears, once you buy that upgrade.

The treasury itself is a chest in the northeast side of the room.

Solution 2:

The Treasury should be just inside the main hall (the building you went into the first time you visited Caed Nua), on the right hand side. If you walk in and down the hall, you should see a room off to the side at the bottom of your screen. If you reach The Steward, you've gone too far.

There'll be a chest inside with your stuff.

Solution 3:

The Great Hall Treasury Chest is located, as the name suggests, in the Great Hall (marked Caed Nua Keep on Caed Nua overview map).

Location in the Great Hall:

In order for it to be there you need Main Keep stronghold upgrade - though I think that you need that anyway to send people on missions.