So I'm tinkering with Jeskai Eldrazi as a pet deck and I have some quick questions about the intricacies of the cast trigger of Eldrazi Obligator (particularly how it interacts with Stifle).

So let's say I just casted the Obligator and announced the cast trigger:

1) When can my opponent Stifle this? Can they wait until I've payed the ability's cost to Stifle it, or do they have to Stifle it before I pay?

2) If they have to Stifle it before I pay the cost, when am I required to announce target? As part of the trigger going on the stack (I.E. they'll know before they Stifle it) or can I wait until I pay the cost to declare the target?

I've looked around and I'm pretty sure I know the answer to the first one, but I want to double check. I haven't had any luck on the second question though.