If someone had Possibility Storm in play and there was also an Ice cave in play, can you pay 0 to counter the card played since it is reduced to a free cost. Another scenario is if someone played a kicker and overload. Do you also have to pay for kicker costs to counter the spell and the overload cost?
A spell's mana cost (any object's mana cost for that matter) is always what's written on the upper right corner of the card, no matter what was actually paid to cast it, if it was cast at all. In order to counter the spell cast via PS, the player has to pay actually pay what the spell would cost normally even if it was cast for free. For example, to counter a Murder cast via PS, the player has to pay 1BB.
Likewise, alternative and additional costs don't count towards the mana cost of a spell. You only have to pay what's written in the upper right corner of the card. For example, to counter an overloaded Vandalblast only takes R. (Though you cannot overload via PS, since another alternative cost of "without paying the mana cost" is imposed already, and only one alternative cost can be applied to any given spell.) A kicked Rite of Replication would only take 2UU to counter.
A cards mana cost is a printed characteristic of the card. Cost increases or decreases, or paying an alternative casting cost (such as with possibility storm or for its overload cost) does not change the cards mana cost. Additionally, additional costs, such as kicker, no not change the cards mana cost.
So to run through the examples you listed:
If a player casts a Serra Angel off of possibility storm without paying its mana cost, other players can still counter it using ice cave, but they have to pay the full printed mana cost (in this case 3WW).
If a card with kicker is cast (let’s say Burst Lightning), I’ve Caves let’s you counter the spell by paying R
Basically, you have to pay whatever is printed in the top right hand corner of the card you want to counter with Ice Cave, no matter what was actually paid to cast the card.
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Likewise, alternative and additional costs don't count towards the mana cost of a spell. You only have to pay what's written in the upper right corner of the card. For example, to counter an overloaded Vandalblast only takes R. (Though you cannot overload via PS, since another alternative cost of "without paying the mana cost" is imposed already, and only one alternative cost can be applied to any given spell.) A kicked Rite of Replication would only take 2UU to counter.
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So to run through the examples you listed:
If a player casts a Serra Angel off of possibility storm without paying its mana cost, other players can still counter it using ice cave, but they have to pay the full printed mana cost (in this case 3WW).
If a card with kicker is cast (let’s say Burst Lightning), I’ve Caves let’s you counter the spell by paying R
Basically, you have to pay whatever is printed in the top right hand corner of the card you want to counter with Ice Cave, no matter what was actually paid to cast the card.