No, it would not be against the rules to keep playing spells to increase the storm count and advancing the game state to draw out the clock.
The main reason this is true is because you are still advancing the game state, and there is no way to know you even have a grapeshot in hand so you could easily be drawing for it.
Also, there is no way to know that you really don't have any win con left, so as long as you play at a reasonable pace and continue to advance the game state, you are still following the rules
Your friend is correct.
What happens when you cast a creature with an Equilibrium on the battlefield is that the Equilibrium triggers once, and when the trigger resolves it asks for a payment of a single colorless mana.
If you pay the mana, you can bounce the creature you targeted. If you don't, then you can't.
It's a yes/no option, not something you can pay more than once per creature cast.
Prototype Portal Golem's Heart Door of Destinies
1.) No you would not, as Golem's Heart only triggers when you cast an artifact spell.
The portal is just placing a copy of the Heart on the battlefield, not casting anything.
2.) No, for the same reason.
The door triggers only when a spell of the chosen creature type is cast, so no matter how many Myr copies you make you are not casting any spells, just placing copies of a card on the battlefield.
Each copy is a separate object and all their bonuses are cumulative. So if you had 2 Door of Destinies out and cast a Myr then each one would get a counter and the Myr would get +1/+1 from each Door.
You appear to have completely misread the question.
1.) Abilities exist completely separate from their source. Destroying the enchantment doesn't stop the abilities that are already on the stack.
Think of it like this; I throw a grenade at you, then you shoot me. My grenade doesn't just disappear in midair.
2.)If the creature dies, then it will do what any creature that dies does (barring any effects that say otherwise): Go to it's owners graveyard
No.
Since you can only activate planeswalker loyalty abilities at sorcery speed, there is no way for your opponent to have a spell on the stack that you can counter with a counterspell you grab with Jace's ultimate
So basically what it is brought down to, is their is no way to stop an activated ability like this without a cancel or something else that specifically stops an activated ability correct? Whether it's paying mana for the ability, or tapping it to active the ability, it will still resolve?
Yep, basically correct.
Do note that abilities aren't spells, so Cancel will do nothing.
You will need a Trickbind, or another "counter target ability" card
The reason your conclusion is wrong is because abilities actually exist independent of their sources (with a precious few exceptions, which are all spelled out on the card).
This is for the same reason that if I throw a grenade at you, and you shoot me afterwards, the grenade doesn't disappear in midair.
112.7a Once activated or triggered, an ability exists on the stack independently of its source. Destruction or removal of the source after that time won't affect the ability.
Edit in response to post below mine:
You are slightly incorrect. The ability will resolve, it's not countered. It just won't do any damage.
It depends on your language.
If you simply said "Do you want to ID instead of playing?" then he was very wrong in what he did, and you can and should complain.
If you offered anything in exchange for the ID, then it was illegal.
2.4 Conceding or Intentionally Drawing Games or Matches
If a game or match is not completed, players may concede or mutually agree to a draw in that game or match. A
match is considered complete once the result slip is filled out or, if match slips are not being used, a player leaves
the table after game play is finished. Until that point, either player may concede to or draw with the other, though
if the conceding player won a game in the match, the match must be reported as 2-1. Intentional draws are always
reported as 0-0-3.
Players may not agree to a concession or draw in exchange for any reward or incentive. Doing so will be
considered Bribery (see section 5.2).
If a player refuses to play, it is assumed that he or she has conceded the match.
The main reason this is true is because you are still advancing the game state, and there is no way to know you even have a grapeshot in hand so you could easily be drawing for it.
Also, there is no way to know that you really don't have any win con left, so as long as you play at a reasonable pace and continue to advance the game state, you are still following the rules
What happens when you cast a creature with an Equilibrium on the battlefield is that the Equilibrium triggers once, and when the trigger resolves it asks for a payment of a single colorless mana.
If you pay the mana, you can bounce the creature you targeted. If you don't, then you can't.
It's a yes/no option, not something you can pay more than once per creature cast.
Golem's Heart
Door of Destinies
1.) No you would not, as Golem's Heart only triggers when you cast an artifact spell.
The portal is just placing a copy of the Heart on the battlefield, not casting anything.
2.) No, for the same reason.
The door triggers only when a spell of the chosen creature type is cast, so no matter how many Myr copies you make you are not casting any spells, just placing copies of a card on the battlefield.
You appear to have completely misread the question.
Think of it like this; I throw a grenade at you, then you shoot me. My grenade doesn't just disappear in midair.
2.)If the creature dies, then it will do what any creature that dies does (barring any effects that say otherwise): Go to it's owners graveyard
Since you can only activate planeswalker loyalty abilities at sorcery speed, there is no way for your opponent to have a spell on the stack that you can counter with a counterspell you grab with Jace's ultimate
If you don't have 6 Apostles to sac, you cannot activate the ability
Since it entered the battlefield under his control, he is the default controller.
Yep, basically correct.
Do note that abilities aren't spells, so Cancel will do nothing.
You will need a Trickbind, or another "counter target ability" card
This is for the same reason that if I throw a grenade at you, and you shoot me afterwards, the grenade doesn't disappear in midair.
Edit in response to post below mine:
You are slightly incorrect. The ability will resolve, it's not countered. It just won't do any damage.
If you simply said "Do you want to ID instead of playing?" then he was very wrong in what he did, and you can and should complain.
If you offered anything in exchange for the ID, then it was illegal.
Here is the relevant section from the Infraction Procedure Guide
Like you said, you're not casting it, just putting it onto the battlefield from your hand.
Can't always overrides can