I think for #1 and #3, as a prediction for the weather later that day, I'd be more likely to use the future tense. As a comment on the weather as it has been so far today, #2 feels more formal and sophisticated than #4.

All are correct. It just depends what you want to say. #3 is something I would say early in the morning after glancing at the weather forecast. #4 is something I would say in the evening. #1 and #2 are certainly correct but "changeable" is not a natural word to use. Better words would be: "variable", "capricious", "erratic"

Hey Alla! Great question, I get this one frequently from speakers of Slavic languages. I agree with Simone, but I would add that I don't think a native speaker (at least from North America) would actually say any of these. In my opinion, it would be far more likely for us to say something like "the weather is (so) unpredictable today" or "the weather keeps changing today" or "the weather has changed a lot today." I should add that "a lot" is more informal, and this is something to think about. I hope that helps!

The weather has been changing all day long.