If the target creature or planeswalker is an illegal target as Tundra Fumarole tries to resolve, it won’t resolve and none of its effects will happen. You won’t add mana.

If you cast Tundra Fumarole without spending snow mana, you won’t add any mana. Damage will still be dealt.

If Tundra Fumarole is copied, no snow mana was spent to cast the copy, so you won’t add any mana. Damage will still be dealt.

Snow is a supertype, not a card type. It has no rules meaning or function by itself, but spells and abilities may refer to it.

The {S} symbol is a generic mana symbol. It represents a cost that can be paid by one mana that was produced by a snow source. That mana can be any color or colorless.

Snow isn’t a type of mana. If an effect says you may spend mana as though it were any type, you can’t pay for {S} using mana that wasn’t produced by a snow source.

Some cards have additional effects for each {S} spent to cast them. You can cast these spells even if you don’t spend any snow mana to cast them; their additional effects simply won’t do anything.

The Kaldheim set doesn’t have any cards with mana costs that include {S}, but some previous sets do. If an effect says such a spell costs {1} less to cast, that reduction doesn’t apply to any {S} costs. This is also true for activated abilities that include {S} in their activation costs and effects that reduce those costs.