>>Found a nice example for the latter in Morton-Benson's dictionary. Asked "what is the weather", nobody responds with a definition ("it's a natural phenomenon studied in meteorology", or any other "it is a ..."), but with quality ("nice", "sunny", "rainy", "windy" etc). So, I agree to conclude that both usages are regular.
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>I've honestly not heard "What's the weather". Normally I hear "What's it like out" or "What's the weather like". Or maybe "Is it raining".

I have never met these gents, Morton and Benson, and have no clue why would they put such an example in their dictionary. But it's there - probably picked up from a book from another era. Dictionaries are notoriously obsolete.

BTW, I was digging a bit and sure enough, there's a corresponding pair of interrogative pronouns in Latin: compare quis with cui at http://www.orbilat.com/Languages/Latin/Grammar/Latin-Pronouns_Interrogative_and_Relative.html