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If an ability would produce one or more mana of an undefined type, it produces no mana instead.

Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.

To “tap [a permanent] for mana” is to activate a mana ability of that permanent that includes the {T} symbol in its activation cost. See rule 605, “Mana Abilities.”

An ability that triggers whenever a permanent “is tapped for mana” or is tapped for mana of a specified type triggers whenever such a mana ability resolves and produces mana or the specified type of mana.

A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.

If an object’s enters-the-battlefield triggered ability or replacement effect refers to X, and the spell that became that object as it resolved had a value of X chosen for any of its costs, the value of X for that ability is the same as the value of X for that spell, although the value of X for that permanent is 0. This is an exception to rule 107.3i.

The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The untap symbol is {Q}. The untap symbol in an activation cost means “Untap this permanent.” A permanent that’s already untapped can’t be untapped again to pay the cost. A creature’s activated ability with the untap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The first ability causes Sarkhan to deal damage directly to himself, resulting in that many loyalty counters being removed from him. The damage would never have been dealt to you, so prevention effects that would prevent damage from being dealt to you won't affect it.

If the targeted creature is an illegal target by the time the second ability resolves, the ability doesn't resolve. The player won't get a Dragon token.

If you target an opponent with the third ability, you choose separately for each Dragon creature you control whether or not to redirect the damage dealt by that Dragon to a planeswalker that player controls.

You choose a single target as you activate the third ability. All your Dragons deal damage to that player.

If the targeted creature is a legal target by the time the second ability resolves but its controller can't sacrifice it (due to Tajuru Preserver, perhaps), the ability continues to resolve. That player gets a Dragon token.

Unlike previous planeswalkers, Sarkhan the Mad has no way to raise his loyalty.

Once the first ability starts to resolve and you reveal the top card of your library (and see how much damage Sarkhan the Mad deals to himself), it's too late to respond.