The reason I ask is for this hypothetical (yet commonly occurring) situation. I am playing emrakul oath. I oath up an emrakul and it is somehow disabled, but still in play (detained or something). I oath again and dragon breath flips, followed by emrakul #2.

Does the legend rule kick in before dragon breath's trigger hits the stack? That is, will the sacrificed emrakul trigger stack first, then DB, thus returning DB to the new emrakul before shuffle? As I understand, the cards hit the grave and battlefield simulatneously off of oath. What happens?

Similarly, what if DB is already attached to the first emrakul when i oath and flip E-money #2?

You can only put it on the one that stays in play after the legend rule since you will have to chose which dies before you can put the triggered ability for Dragon Breath on the stack. If Dragon Breath is already attached then it won't have been in the graveyard at the time that the Second Emrakul entered the battlefield so in that case Dragon Breath won't trigger at all.

Emrakul is oathed into play
Dragon Breath triggers
You choose an Emrakul
Other Emrakul dies and shuffles your yard
Dragon breath trigger fizzles since its no longer in the GY.

The legendary rule is a state-based action that happens anytime a player would get priority, before they get priority. It has the same "speed" as other state-based actions like a creature or player dying to lethal damage or a token being annihilated for existing in the wrong zone.

If you Oath into an Emrakul, flipping Dragon Breath along the way, you will get a Dragon Breath trigger regardless of other Emrakuls. The trigger will actually go on the stack before the Legendary rule takes effect, but the Legendary rule will take effect before any trigger resolves and before any player can use priority to cast spells or activate abilities.

If you have an Emrakul wearing Dragon Breath and Oath into a second Emrakul the Legend rule does not take effect until a player would gain priority, so the window for triggers to go on the stack (as a result of resolving the Oath ability) has already passed. The second Emrakul cannot gain the Dragon Breath from the first as a result of the normal and obvious triggers (obviously you could blink Emrakul with a Venser planeswalker or something so it is possible for it to get the Dragon Breath through some later more complicated route)

@vaughnbros that's a good point, the shuffle trigger will go on the stack above the Dragon Breath trigger and prevent it from resolving (note that it is a trigger though so you could respond to it, and it could potentially interact with weird effects like Samurai of the Pale Curtain)

State-based effects are checked before triggers are placed on the stack. The proper sequence is:

1. Oath resolves putting Dragon's Breath in the graveyard and Emrakul into play at the same time.
2. Dragon's Breath triggers but is not yet put on the stack.
3. State-based effects are checked and the controller of the Emrakul's chooses one to keep and sacrifices the other.
4. Emrakul's leaves play ability triggers but is not yet put on the stack.
5. The active player puts their triggers on the stack in the order they choose - they should choose to put the Emrakul ability on the stack first and Dragon's Breath second.
6. Abilities resolve in the reverse order - Dragon's Breath enters play before the shuffle effect.

Thank you @DBatterskull for confirming this (perhaps not the proper terminology and explanation but that the player will choose the order in which the triggers occur).

@ChubbyRain it looks like you're right, but I think MTGO handles it differently