Because it's a loyalty ability, Narset's first ability isn't a mana ability. It can be activated only any time you could cast a sorcery. It uses the stack and can be responded to.

Narset's second ability goes on the stack without a target. While that ability is resolving, you draw a card and may choose to discard a card. When you discard a nonland card this way, the reflexive triggered ability triggers and you pick a target to be dealt damage. This is different from effects that say "If you do . . ." in that you choose a target after looking at the card you draw and players can respond to the reflexive triggered ability knowing how much damage will be dealt.

You can discard a card even if Narset leaves the battlefield before her middle ability resolves. The reflexive triggered ability will still trigger if you discard a nonland card.

If a card you discard has {X} in its mana cost, X is considered to be 0.

An ability that triggers when a player casts a spell resolves before the spell that caused it to trigger. It resolves even if that spell is countered.

The emblem created by Narset's last ability is colorless. The damage it deals is from a colorless source.