As a native speaker, this is a perfectly common type of phrase. It looks very strange in English because we don't speak that way, but in Russian it's a common way to emphasize the question to be asked.
You are first addressing the people or thing you are referring to, then interjecting with the question word, before continuing with the rest of your statement. Hope that helps a bit.
No, it's not quite like that. It's more just emphasising the question - it kind of makes it sound more incredulous. As for your example, the correct way to phrase that would be, 'ты что, разбил окно?' (without the 'скажи' sounds better to me) This question implies some level of shock at seeing a broken window, without much other context (+ someone else is also there, obviously).
Could someone explain this to me?
"What, we do not have milk?" To me this sounds like anger/sadness at finding out that there is no milk left for the morning breakfast cornflakes or something like that. Does that fit?
"У нас что, нет молока." To me this looks like the что is completely out of place. I don't understand this construction.
To those who are confused about the English translation or say it's not correct: We are not learning English, we're learning Russian and I think we all need to learn to FORGET how we express things in our native language and learn how people in Russia (and other Russian speaking nations) express themselves.