Does every dungeon have a quest that would have eventually led you to that dungeon ? If you cleared it before getting the quest does that interfere with the possibility of 'correctly' completing the dungeon ?
Thanks for any info, sorry if this has already been discussed.

Not every dungeon has a quest. Some dungeons have little side stories to go along with them but some are just plain old dungeons. As for your final question, the only dungeon I can think off the top of my head that you shouldn't enter before starting a quest is Temple of the Ancestor Moths. If you're playing on PC I'd suggest downloading the unofficial patches, this should fix any issues with quests and dungeons.