Whenever a quest instructs you to do a dungeon, you never have to do anything special in the dungeon to complete the quest. You just have to complete the dungeon. You will get quest credit either immediately upon boss death, or upon exiting after boss death. This holds true for quests that want you to obtain something: the required item will be added to your inventory automatically when you exit.


In re: Doom, cleanse it by stepping on the glowing tile. Make sure the status disappears from your bar. The light moves around, and you might step on the tile just as the glow moves to another tile, so don't stop unless you see it cleansed.