They're both correct. The difference is that the second is asking for a more precise figure, and so it sounds more formal/scientific/academic. Because of this, the second might not sound so natural in casual contexts. For example, if you're chatting with a friend about a river that you've visited, your friend would probably ask "How long is the river?", and you could give an approximate answer. "What is the length of the river?" sounds like something out of a geography test, and it would require an exact figure as an answer. The same goes for other measurements. In everyday situations, you'd say "How tall are you?", whereas "What is your height?" is something that you'd encounter on a form you might fill in for a medical examination.

I would say the first is more casual and the second is maybe a bit more technical. I would tend to maybe use the first in spoken English and perhaps the second in written English. But, both are correct.

Hi! Both these sentences sound completely fine :) And you can of course do it with weight, breadth, height, width and depth. How deep is the river? What is the depth of the river? Same thing :) The only minor difference is that when you are using the nouns (depth, height, etc) it sounds a little bit more formal, but I wouldn't really dwell on it :)