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http://mabination.com/threads/74946-Mabinogi-Project-Dream-Treasure-Hunter-Update-%28Consolidated-and-Orginized%29

So, according to the above:

• You need at least Rank 6 Exploration Mastery and/or Treasure Hunter as active talent to find a Treasure Dungeon.
  • Treasure dungeons shares its spots with any relic that can be found with L-Rod Exploration.
  • When you find one you get a prompt that asks if you want to unearth the treasure dungeon, otherwise you'll unearth the relic instead.
• You can't go in while in a party, whether or not you're leading.
• Other people can't go in your treasure dungeon.
• Once you use the dungeon, it disappears.
• The dungeon always appears as Longa Desert regardless of where you find it.
• The dungeon consists of three rooms: two puzzles with five emotional faces and the boss.
  • The altar gives you some sort of "describe me" riddle and you have to hit the right statue related to it.
    • Hitting the right statue gives extra chests at the end of the dungeon.
    • It doesn't specifically say what happens when you get it wrong. I'm guessing you just fail the puzzle and you just move on.
  • Boss is a Stone Mask with higher damage and health, and can summon other Stone Masks and Ghost Masks.

I have not noticed extra chests, unless you mean hitting the right statue in the second room.

However I have noticed that when I do the first room "correctly" I get a box plus a suntouched material.

I dunno, I just pulled it off from Mabination. I have yet to do the dungeon myself (though I don't have any access to it at the moment).

The "extra chests" may have to do with the specific Ancient Reward Chests you get at the end and not the reward chest you open on the ground to get the Ancient Reward Chests. However I've only ever found 1 or 3 and the JP wiki supposedly has someone saying that even though he did it wrong on purpose he still got 3 one time.

From personal experience whether I do the first room right or wrong doesn't seem to affect whether I get a suntouched item or the number of reward chests.

Small sample sizes. I haven't done enough "wrong" to see if I can still get suntouched though. If we can, and the other statements are taken true, then it seems the first room does not make much of a determination in the rewards. IF IT WERE ME, I would make the first room increase the number of "correct" faces in room 2 for a chance at better rewards.

Well on my first attempt I had no idea what I was doing and got three Ancient Reward Chests and a Suntouched Stone Fragment. Regardless of sample size the probability that I got it correct is very slim so there's that to consider.

the probability you got it randomly right is 1/120, so definitely feasible.