×2Snow CoatingDouble the number of damage counters on each of your opponent's Pokémon.
BlizzardThis attack also does 10 damage to each of your opponent's Benched Pokémon. (Don't apply Weakness and Resistance for Benched Pokémon.)
f both heads get angry simultaneously, this Pokemon expels a blizzard from its horn, burying everything in snow.
If there is a Battle Cage in play, and I use N's Vanilluxe's "Snow Coating" attack, what happens if my opponent has a Benched Pokemon with 3 damage counters on it?
Doubling the number of damage counters requires damage counters to be placed. So the damage counters would not be added to your opponent's Benched Pokemon; it would be protected by the Battle Cage.
If N's Vanilluxe uses its "Snow Coating" attack on a Pokemon that already has two damage counters on it, does it place four more damage counters on that Pokemon for a total of six?
No, that's not now it works. Snow Coating counts how many damage counters are already on the Pokemon and places the same number of additional damage counters on it. So in your example a Pokemon with two damage counters would now have a total of four on it, not six.
If my opponent's N's Vanilluxe uses its "Blizzard" attack and there is a Battle Cage Stadium in play, would Benched Pokemon be protected from taking 10 damage each?
No. Doing damage is not an effect and is different from placing damage counters. Battle Cage only protects against placing damage counters, not damage done, even if that damage came from the attack text.