Kogarashi Level 0 Judge

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Spending as though it was doesn't change the fact that it's colorless, not red. Adamant would not apply.

From the Gatherer rulings for Slaying Fire:

• Adamant effects check what mana was actually spent to cast a spell. If an effect allows you to spend mana “as though it were mana” of any color or type, that allows you to spend mana you couldn’t otherwise spend, but it doesn’t change what mana you spent to cast the spell. (2019-10-04)

Additionally, colorless is not a color, so you can't spend mana as though it were specifically .

Fell Specter will trigger, but your opponent will lose before the Fell Specter triggers would go on the stack.

A player losing the game due to having 0 life is one of the game's state-based actions (SBAs). These are checked whenever a player would receive priority. Additionally, triggers go on the stack whenever a player would receive priority, and in order for items on the stack to resolve, all players must pass priority without adding anything to the stack (this applies to each item).

What happens in your scenario is that you cast Fateful Showdown. Assuming no responses (which would go on the stack above it), Fateful Showdown then resolves in its entirety. This means that the Showdown deals the damage, which you stated to be lethal, and then you discard your hand and draw new cards. Technically Fell Specter triggers at this point, but the triggers do not go on the stack just yet since Showdown is still resolving.

Once Showdown is done resolving, the active player would receive priority again, so SBAs are checked before triggers are placed on the stack. SBAs see that your opponent is now at 0 or less life, and your opponent immediately loses the game. Since your opponent is gone, so are their triggers, which never get to go on the stack (which matters in a multiplayer game where you still have another opponent).

Yes, as long as you have controlled that creature continuously since the beginning of your turn. Unlike cards that temporarily exile things, like Oblivion Ring, Imprisoned in the Moon doesn't cause the creature it's enchanting to change zones. Once the enchantment is gone, your creature will stop being a land and go back to being a creature again, without summoning sickness unless it only just entered the battlefield this turn.