Is the sentence "Don't you cry" (or don't you speak, don't you cry etc.) is correct? For me, as ELL, it looks weird to see pronoun (in this case "you") in such sentence - as an order rather than a question. I expected it to be: "don't cry" instead of "don't you cry". what is the explanation for that?
Nothing wrong with that. It's just a normal structure of imperative with a visible subject. You can use it to convey a tone of friendly encouragement.
This structure can also be used to express impatience or irritation. It creates an agressive hectoring effect.
"Don't you call me your friend!"
Don't you cry.
The former sentence is a normal imperative negative sentence. The latter can also be treated so as an imperative. They have the same meaning; you are asking the listener, you, not to cry. You are using the pronoun you in the second sentence to emphasize the listener, you.