What is the proper way way to play Butcher of Malakir and Falkenrath Noble?
Scenario 1

As the creatures are dying, more or less.


Butcher of Malakir's ability triggers 6 times. Falkenrath Noble's ability triggers 6 times.
You place the ability on the stack in the order you desire (since they're all yours): [Doesn't really matter]


Butcher of Malakir's ability is put on the stack 6 times.
Falkenrath Noble's ability is put on the stack 6 times targeting whoever.

Falkenrath Noble's ability #6 resolves:


The targeted player loses one life.
You gain one life.

Repeat step #3 for the other 5 instances.
Butcher of Malakir's ability #6 resolves:


The opponent sacrifice one of his 5/5s. Falkenrath Noble's ability triggers.

Falkenrath Noble's ability is placed on the stack targeting whoever.
Falkenrath Noble's ability resolves:


The targeted player loses one life.
You gain one life.

Repeat steps #5-#7 for the other 5 instances.


(If the other opponent has creatures too, there will be more triggers added in step #6.)

So before moving from the Combat Damage Step to the End of Combat Step,


One or more players lost a total 12 life (one at a time).
You gained 12 life (one at a time).
Each opponent sacrificed 6 creatures.


Scenario 2

It's #2 (you take damage) because all combat damage is dealt simultaneously[1].


Simultaneously, 


Your three 1/1s deal one damage to each of his 1/1s.
Three of his 1/1s deal one damage to each of your 1/1s.
The other three of his 1/1s deal one damage each to you.

SBAs are performed simultaneously:


You lose the game if you have non-positive life.
Your 1/1s are destroyed. (Butcher of Malakir's ability triggers three times. Falkenrath Noble triggers three times.)
His three 1/1s with one damage marked are destroyed. (Falkenrath Noble's ability triggers three times.)

You put the nine triggered abilites on the stack in the order you desire.
...




Notes:


All combat damage dealt by the creatures participating in that Combat Damage Step is dealt simultaneously, to be specific. Creatures with First Strike and creatures without participate in different Combat Damage Steps.