#casting triggers

[magicjudge]

Does Meren's ability count as 'casting' for cards like Ulamog, the Ceaseless Hunger?

Putting a creature directly onto the battlefield isn’t the same as casting it.

[magicjudge]

If I Æthersnatch an opponent's Ulamog, the Ceasless Hunger, do I get to exile 2 permanents?

No.

Ulamog’s casting trigger exists independent of the Ulamog spell. 

Your opponent controls the triggered ability, and Æthersnatch only gives you control of the spell, not the ability that triggered when it was cast.

[magicjudge]

If I cast Aethersnatch in response to my opponent's creature spell that has a "When you cast... do this" like Ulamog, the Ceaseless Hunger's exile two permanents am I able to change my opponents targets?

No.

You’ll gain control of Ulamog, but your opponent still controls the exile trigger since it exists independantly of Æthersnatch.