I've tried using my fighting lupora, that is strictly stat'ed for strength (950), and nothing happened. : \

And for some reason, I've trudged through the dungeon 9 times in a row, and the best gear I've been getting is Epic gear, no legendary or mythical gear what so ever.

After you open the door, you won't be prompted into an immediate battle like the ones in Nightfall Dungeon. You will need to explore the dungeon a few times, then you will get to see the optional dungeon boss. Also, the description on each door determines which pet should be used to open them. If I'm not mistaken there should be 4 types, I will try to describe them and the stat requirement to open each door:

1. The door won't budge = High STR
2. You don't see a way to open them = High INT
3. The door is locked = High DEX
4. A switch high above the door = High AGI


EDIT: Mythical drops are purely luck I'm afraid... I had an okay luck with equipment drops, but not so much on avatar item drops. DX

The lowest stat I have used to open the door is...419. So, I'm assuming you might need around 400 of the required stat to open the door.

I'm gonna chip in that Lehana's guess is probably correct, because I tried opening the Agi door with a 339 Agi pet, and he was unable to open it. x.x