If a permanent enters the battlefield as a copy of Sarkhan after his first ability has resolved, the copy will be a Sarkhan planeswalker. It will enter the battlefield with four loyalty counters, no matter how many loyalty counters are on the original Sarkhan. You'll then find yourself in the same situation described above.

If Sarkhan, the Dragonspeaker becomes a creature due to his first ability, that doesn't count as having a creature enter the battlefield. Sarkhan was already on the battlefield; he only changed his types. Abilities that trigger whenever a creature enters the battlefield won't trigger.

Any loyalty counters on Sarkhan when his first ability resolves will remain on him while he's a creature. Because he's not a planeswalker at this time, damage dealt to him won't cause those counters to be removed.