Just out of curiosity...
If I use his ability to turn into a 4/4 dragon (with indestructible/haste), could I create a permanent copy of the dragon using clone?
No. When copying objects, you copy the card as printed (except when applying effects from other copy effects). So, in this case, you would end up with an unanimated Sarkhan and you would end up having to put one in the graveyard at the end of the turn when the original Sarkhan turns back into a planeswalker.
To clarify: Sarkhan's first ability doesn't generate a copy effect. Rather there are four different kinds of effects: a type-changing effect ("legendary ... Dragon creature") (C.R. 613.1d), a color-changing effect ("red") (C.R. 613.1e), an effect that adds abilities ("with flying, indestructible, and haste") (C.R. 613.1f), and one that changes power and toughness ("4/4") (C.R. 613.1g).
With the release of Ixalan, Sarkhan, the Dragonspeaker is now legendary (even outside the battlefield and even if Sarkhan's first ability hasn't resolved yet). That means if you control Sarkhan and you bring Clone to the battlefield as a copy of it, you will control two legendary permanents named Sarkhan, the Dragonspeaker, so that (due to the legend rule) you will have to have one of those permanents go to the graveyard as a state-based action — which happens the next time a player would get priority (C.R. 704.3, 704.5j).
Not completely accurate. There are ways to either create non-legendary token copies (Spark Double, for instance) or even negate the "legendary rule", like Cadric, Soul Kindler. Because neither actively violates the ruling, they are valid examples of creating copies without the need to destroy one or the other.
Mind you, these are merely examples that allow for more than one, not actively suggesting to use them specifically.
Spam warning issued. Please do not revive 7-year old threads just to bring up the fact that specific cards can change the scenario asked about.
If I use his ability to turn into a 4/4 dragon (with indestructible/haste), could I create a permanent copy of the dragon using clone?
Sarkhan, the Dragonspeaker
No. When copying objects, you copy the card as printed (except when applying effects from other copy effects). So, in this case, you would end up with an unanimated Sarkhan and you would end up having to put one in the graveyard at the end of the turn when the original Sarkhan turns back into a planeswalker.
EDIT (Mar. 26, 2019): Clarification.
EDIT (Aug. 28, 2020): Change rule citation; correctness edit.
EDIT (Apr. 28, 2021): Edited.
Mind you, these are merely examples that allow for more than one, not actively suggesting to use them specifically.
Spam warning issued. Please do not revive 7-year old threads just to bring up the fact that specific cards can change the scenario asked about.