[Card Interaction]

[Raider Bob]

Planar Void Trigger with Split Second.

Situation Nimble Mongoose hits the Yard with a Planar Void in Play, in response to the planar void trigger my opponent Extirpates Nimble Mongoose, Extirpate Resolves, Planar Void Triggers for Extirpate can I Extirpate his Extirpates or is split second from the first extirpate still on the stack?

Basically when does split second leave the stack to play activated abilities or other spells?

[Nightmare]

Planar Void Trigger with Split Second.

Situation Nimble Mongoose hits the Yard with a Planar Void in Play, in response to the planar void trigger my opponent Extirpates Nimble Mongoose, Extirpate Resolves, Planar Void Triggers for Extirpate can I Extirpate his Extirpates or is split second from the first extirpate still on the stack?

Basically when does split second leave the stack to play activated abilities or other spells?
Split Second isn't a triggered ability - it doesn't use the stack (so it can't be responded to). Here's the way that interaction occurs:

Planar Void in play.

Your Nimble Mongoose dies, say from combat damage. It is placed into the graveyard when SBE's are checked. This triggers the Planar Void, and in response, your opponent plays Extirpate on the Goose.

At this point, you cannot respond to the Extirpate on the Goose.

Extirpate resolves, and he extracts your Geese. As the final part of resolution, the Extirpate is placed in the graveyard, which triggers the Void.

Here is the reminder text of Split-Second, emphasis mine:

As long as this spell is on the stack, players can't play spells or activated abilities that aren't mana abilities.
Once the Mongeese are removed, and the Extirpate triggers the Void, the spell is resolved, and has been taken off the stack. You are free to respond to the Planar Void trigger, and Extirpate his Extirpates. (Note that he will be able to respond to the Void trigger of your Extirpate, as well.)