I know that in WOW, if you dont pick up quests, and start exploring, it usually means that you can't pick up an important quest/item loot, or you don't get credit. As a result the time can feel wasted.

Is there any danger in Skyrim of completing dungeons you happen to pass on the road?

No. But if you get a quest to go to the same dungeon that you would have went to without the quest, then you're getting more rewards.

A lot of quests are just to give you a reason/motivation to clear dungeons.

Once, you've cleared the dungeon, does the quest go away (questgiver won't ask you to do it), or do you end up getting a quest stub, that gets completed immediately because you already have the quest item?

You can pick up quest items in dungeons without the quest. But if you do your stuck with the item in your inventory untill you actually find the quest, and even then it usually bugs out and you can't complete the quest anyways. Great game, but there is so many quest bugs. I really hope they are working on fixing them. i just hit lvl 50 and have about 40 lbs of quest items that i can't get rid of because of bugged quests. Very annoying

Yes, you can. More specifically, you can break quests you do not have yet.

Doubt it. I managed to get the Elder Scroll before even talking to the Greybeards, and my main story was fine in the end.

I'm having a similar experience to you, but not to that extent... 40lbs, that's nuts.

I have one quest for the Jarl of Riften to clear out a skooma den. I cleared it out before getting the quest and it's not updating at all. So possibly.

OP,

I created a similar thread here;
http://www.gamesas.com/index.php?/topic/1296060-random-exploring-may-cause-problems/

On page two, I confirmed a working "work around" that another poster provided.
This pertains to Quest Items gathered prior to a Quest.