I was just wondering... Is it possible to clear every dungeon without joining all the factions? I know Saarthal is the first major dungeon for the College of Winterhold, but is it possible to complete it without joining the College?

No, I don't think so.
Sarthaal isn't the only dungeon that the Mages college give you access to either.

You don't need any factions to complete a dungeon. Just go in there and wail away. Or sneak.

However, what will bite you in the butt is when the Faction sends you back to a Dungeon for some reason, the NPCs will be back and will be much tougher to deal with if you cleared it out already. To that end, even if you clear out a dungeon, the denizens return over time. And, they are stronger when they do.

So, I go into some dungeons while exploring, and some I don't. The game then tends to send me places that I have not been yet.

I think the OP wants to know specifically about Saarthal (which requires a key to gain access), probably pertaining to the Forbidden Legend quest. I've read that the main story line takes you to the College at some point, haven't hit that point in the "main" storyline (even after 200+ hours). I had to join the College to get in to get the fragment, that's the only way I know how.
And I believe that the Thieves Guild end quest inside the Dwemer ruin can really only be done as part of the Guild quest line, but don't quote me on that!

The answer is no, some dungeons are quest specific and locked off until you have the related quest, some of those quests are faction specific. I'd say about 95% of the dungeons are free to roam but ~5% are either fully or partially locked off until you have the related quest. One of the dungeons literally despawns rubble that blocks off a route when you start the related quest from the Civil War Campaigns.

Saarthal itself is locked to the mage's guild, I got lucky on my first play through since after finding it locked I decided to go to the college of winterhold for the mage's guild questline...

Pretty much just.. blatantly, patently, obviously false. There are some dungeons which can not be accessed without activating certain quests.

Have you played the game?

I highly doubt it.

Some doors need "special" keys to unlock.