Did Nameless King Kill Ornstein?

In any case, Ornstein eventually finds his way to Archdragon Peak. The fact that Ornstein's equipment isn't pillaged from a corpse seems significant. It suggests that the Nameless King did not fight and kill Ornstein (or else we should expect to find his body).

Was Ornstein killed by the nameless king?

I don't think the Nameless King is Faraam, and that is simply a theory rather than an actual fact in the game. But true, he did not kill Ornstein it seems.

Did Ornstein die?

After killing ornstein his people found his body and brought him back to archdragon peak. food for thought: maybe the dragonslayer armor we find there isn't ornstein's, it's the nameless king's.