Very interesting and worthy topic, Kibernal!

I am definitely a proponent of the idea that Aulë instructed the Khazad in their language. If you reference the Silmarillion, you see a very interesting spiritual situation arising; the Dwarves are the only living things, talking or non talking, which are not by some means the Children of Illúvatar. Even Melkor, Sauron, Orcs and Evil Men all had their origins in the thought of Illúvatar and the Music of the Ainur.

Aulë is the father of the Khazad. He loved them dearly, and although he was willing to submit to the will of Illúvatar and destroy them if that was his wish, he gladly set the fathers of the Khazad to sleep.

The Silmarillion states that the Khazad have their own beliefs about where their souls go; that they are gathered in separate areas of the Halls of Mandos, and that Aulë gathers them to him.

This belief is as near to deity worship that we can really see. This is the first indicator that the Khazad have a very close and special relationship with Aulë.

The second big indicator is that the Dwarves are incredibly secretive about their language and their names. I believe that this comes from the fact that their very creation was a secret. I think that either as the fathers of the Khazad slept, or shortly after they awoke, Aulë taught them, knowing that Illúvatar had agreed to look the other way about creations that could be considered, technically, abominations.

The fact that true names are concealed, and language protected, would indicate to me that they hold these things very dear and don't want to share them with just anybody, especially the Children of Illúvatar. The Dwarves are a people completely unique from every other thing, down to the atomic level. Their identity and their meaning for existence is tied to Aulë. By embracing him as their father/diety, they gain not only pride in being his only people, but a means of not feeling like outcasts in the world. By embracing their origin with Aulë, they can claim being the first people to be created. True, the Elves were the first to officially be ushered into the world, but it does not change the fact that the Dwarves were created first. That alone is enough for the Khazad to redeem a substantial amount of pride in their race.

I believe that Aulë continued to teach the Khazad, in secret places, which would also explain their affinity for deep and secure homes. (Of course, this also is tied in with the idea that they were made from the mountains, and slept within them for so long.) I believe that he taught them skills which no others would ever master.

While all of the other races are described as being in a state of decay and dwindling, the Dwarves seem to be improving as time goes on; what was probably viewed as an abomination earlier could now easily be considered a fortunate accident, even a blessing. In allowing Aulë to make amends for his naïve mistake, Illúvatar permitted him to become more like himself than any of the other Ainur.