Dear EW:

You wrote:
> -“He didn’t look English.”
> -“What did he look?”
> -“He looked Armenian.”
>
> Is the second sentence in this imaginary conversation correct and natural?
> If it isn’t, what would one say instead of it?
>
Anyone who used the second sentence would betray himself as a non-native speaker. It probably would be understood, but a question with “look” as a linking verb normally has a qualifying adverb with it.

In this case you would say, “What did he look like?” In many cases, people would say “He looked like an Armenian” anyway.

If you were inquiring about a person’s health or physical condition, then (“He looks healthy”) you would say, “How does he look?”

I hope this helps.