I was given a quest of going to the Tower of Ghirtt and picking up a scroll. The quest giver told me to not open the scroll.

In the tower, I found a bag. Something popped up on the screen really quick but I didn't see it before it went away. The bag has a green background on my inventory list. Does this mean I have the item that I was sent for?

When I went back to the mage's guild, the quest giver isn't letting me give him the bag nor will he give me a new quest. The quest is still showing on my quest log. Is it just a buggy quest or do I still need to keep searching for the item?

The problem resolved itself. I had decided to leave Totambu and head back to Wayrest in hopes of getting the royals to cough up more info. When I arrived in Wayrest, a dialog box popped up thanking me for finishing the quest and gave me the quest reward which was some magical shoes. Very weird.

Usually when there is something you aren't supposed to open, you can change the quest by opening it. It's usually something interesting to read too, if it's a letter. Think of other people who would want the info, then deliver it to them instead.

That quest is from a quest pack and, like most quests in that pack, is bugged. I suggest you stick with the original quest pool, and don't bother with the patches and whatnot.