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.

The check of whether Replicating Ring has eight or more night counters on it happens only as part of the resolution of the triggered ability. If Replicating Ring picks up its eighth night counter at another time, you won't immediately create Replicated Rings. You'll have to wait until the next time the triggered ability resolves.

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

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.