The Stack

Nivmagus Elemental and guttersnipe on the field. If I cast any instant/sorcery does the 2 damage happen if I choose to exile it after casting it to gain the 2 1/1 counters?

I would say you would get the 2 damage. Cast the spell, 2 damage, then exile before it resolves to pump up the elemental. I could be wrong tho . . . Not 100% sure.

I surfed the MtG site and YES you get the two damage AND can exile it for the 2 counters.

When you cast your instant/sorcery, the Guttersnipe's trigger will go on The Stack on top of it. Once it is there, it doesn't matter what happens to the original spell, the 2 damage will still resolve as normal.

Situation:

My turn, I have a  Helldozer on the BF and 7 swamps

My opponent has a  Ballynock Trapper and 4 lands, 2 of which are non basic

I pay 3 and tap the  Helldozer targeting one of the non basic lands

My opponent in response activates the tap ability of the  Ballynock Trapper

He believes that the  Helldozer would be tapped after his ability resolved because of his Trapper's ability

My belief is that the  Ballynock Trapper taps my already tapped  Helldozer, because of APNAP, then the dozer's ability resolves - the land would be destroyed and  Helldozer would untap

Situation:

My turn, I have a  Helldozer on the BF and 7 swamps

My opponent has a  Ballynock Trapper and 4 lands, 2 of which are non basic

I pay 3 and tap the  Helldozer targeting one of the non basic lands

My opponent in response activates the tap ability of the  Ballynock Trapper

He believes that the  Helldozer would be tapped after his ability resolved because of his Trapper's ability

My belief is that the  Ballynock Trapper taps my already tapped  Helldozer, because of APNAP, then the dozer's ability resolves - the land would be destroyed and  Helldozer would untap

You are correct you'll be able to destroy 3 of his lands.

I also tried to explain to him, that if he waited to tap the  Helldozer until after the Dozer's ability resolved that I would get the chance to respond which would be tap 3 destroy another land and untap. Helldozer would then be tapped by his creature's tap ability.

Assuming I have the mana to pay for it, would I be allowed to activate the  Helldozer's ability inbetween the resolution of Dozer's previously activated ability and the  Ballynock Trapper ability as the stack was resolving as a response?

You're correct Bilygote. And yes, you may respond to the resolution of Dozer's ability, and assuming all criteria are met to pay for it (dozer is untapped and you have the mana) you can use dozer's ability again, adding it to stack on top, leaving your opponent's ability sitting patiently on that stack waiting until you're done.

Thanks

^ bouncing a creature targeted for destruction still works though right? It would make his spell fizzle?

Also.. What decides when the current stack ends? Just when both players agree? So suppose i played a split second spell, a new stack would jus start?

^ bouncing a creature targeted for destruction still works though right? It would make his spell fizzle?

Yes; the key word is target. If you target something, but in response someone removed that thing from play, the targetting spell or ability would fizzle. Which brings us to...the stack.

Also.. What decides when the current stack ends? Just when both players agree? So suppose i played a split second spell, a new stack would jus start?

Okay, here is an analogy for the stack that will hopefully answer your questions.
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The stack is like a plate, and spells are pancakes. Casting a spell is like placing a pancake on the plate (stack). And eating a pancake is resolving a spell. BUT WAIT. There's a catch! You cannot eat a pancake if another one is on top of it. When you place a pancake on the stack, you let the other person get a chance to place one. Once no player has any more pancakes to place on the stack, it starts to get eaten FROM THE TOP DOWN! Here is a visualization:

So say the plate (stack) is empty. But I decide to cast Thought Scour. When I do this, I place one pancake (spell) on the stack. So the Thought Scour pancake is ready to be eaten (resolved). But oh no, before you eat your thought scour pancake, your buddy decides to cast a Think Twice pancake in response!

So now let's look at the plate (stack). On the bottom is your thought scour pancake. Sitting on top is the Think Twice pancake. Remember, the stack must be eaten (resolved) from the top down. No exceptions!

So Think Twice is eaten first! This means the my opponent draws a card. Once he does, my Thought Scour is the only pancake left on the stack. But wait a second...my opponent isn't done yet. He decides to place another pancake on the stack: Lightning Bolt

So now there are 2 pancakes again! This is because when there are 2 or more spells on the stack, you can let the first one resolve and then respond to the one that's left! Cool, huh?

So the lightning bolt pancake is eaten and I take 3. There is now once again, one pancake on the stack! My lovely thought scour. Now my opponent has no more responses, so I can eat it. Now I mill myself 2 and draw a card because my thought scour pancake was finally eaten (resolved)!

Now for split second cards. When a split second pancake is on the plate (stack), NO OTHER PANCAKES MAY BE PLACED ON THE PLATE!!! Think of split second cards as nasty pancakes that no other pancakes want to share the stack with. So if you cast a split second spell, NO PLAYERS may cast spells until the split second one resolves. This means it can't be countered, or responded to.
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Hope the whole pancake thing relates. I know I luv pancakes

Holy mother of god sweet infant baby Jesus. This pancake analogy is the GREATEST THING I HAVE EVER HEARD!

MIND=BLOWN

That pancake analogy reminded me of Sexual Harassment Panda's explanation pf sexual harassment laws.. It was also extraordinarily helpful

So suppose i cast vines of vastwood, and it goes on the stack.. He can totally just doom blade my creature making vines of vastwood useless? That seems like a sad life for a card

That is correct. If you are casting multiple buffing spells, cast vines last in case they have a killspell ready.