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.

After the first ability resolves, Sarkhan won’t be a planeswalker until the end of the turn. Later that turn, if you cast another Sarkhan, the Dragonspeaker, neither permanent will be put into its owner’s graveyard. The “legend rule” sees only the original Sarkhan and the “planeswalker uniqueness rule” sees only the new Sarkhan. However, if you activate the first ability of the new Sarkhan, you would then control two legendary permanents with the same name. You’d choose one to remain on the battlefield and the other would be put into its owner’s graveyard. If you don’t activate the first ability of the new Sarkhan, the original Sarkhan will return to being a planeswalker during the cleanup step. You would then control two planeswalkers with the same planeswalker type. You’d choose one to remain on the battlefield and the other would be put into its owner’s graveyard. There would then be another cleanup step before the turn ended.

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.