Okay, so I took a quest from the Mages Guild in Daggerfall city to find and retrieve a dangerous scroll from a dungeon. I found the scroll and returned, but when I go to turn in the quest he gives me a new quest like nothing happened before. I check the log book and see that I'm not late. Then I look in the log book again and in the quest entry it says "The Mages Guild of The Mages Guild" blah blah blah. So, since there's no place called "The Mages Guild," does this mean the quest broke even though I found the quest object?

You took too long.

But just take the next quest... there are no consequences or plot developments involved in guild quests so your not missing anything. They will never advance in story, they will often be copy/pastes of each other give or take the name of a dungeon/item.

The real reward is the loot you aquire dungeon crawling. The 300 gold reward you get from storming an orc infested dungeon is pittance.