Since the movie is not the same as the comics, I'm afraid there is no official answer to this question, as this is not explained in the movie itself. However, using some common sense and knowledge of the Marvel universe, I believe we can reach a good enough guess.
What we see in the movie as Odin "fading away into the wind" is actually Odin transforming himself into pure energy/Odinforce, then making his way into Valhalla to reunite with his wife Frigga. (Idea taken from here, but like I said it does make sense and I can't think of a better explanation.)
In the comics, when Odin dies he pass the Odinforce to Thor. While not explicitly shown in the movie, Thor does become utterly powerful which hints he did get that power nonetheless.
As for the questions in the question:
Asgardians gods are immortal so he didn't die of old age
True, he didn't die of old age, he chose to leave the world of the living. (In the comics he's doing this on purpose to fight Surtur and prevent him from reentering the world.)
He appears to Thor and tells him that "He isn't the God of Hammers.."
While in Valhalla he can still communicate, as he's not really "dead".