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.