Will I be able to get a quest for a dungeon already "cleared"?

Solution 1:

From my experience so far, either certain areas of the dungeon will be inaccessible until you accept the quest or you will be able to complete the quest as soon as it starts by way of having already obtained whatever the dungeon holds in store. So no, you won't prevent any possible quests to that location by clearing out dungeons.

As Dave notes, some quests require you to clear dungeons or kill bosses, in which case if you've already cleared the dungeon new enemies will spawn there that you have to kill.

Solution 2:

there are some areas where you can clear the entire dungeon. When you get a quest to go there, if the quest doesn't involve getting something you may have already gotten, it will repopulate the dungeon, most of the time with enemies scaled to your level