You can say: "what's the weather right now? "

If you learnt "What's the weather like?", then of course you use that phrase. We understand your other phrases, but if you use them, we will hear that your English is not very good.

Whither the weather? also works.

Why not just add to it to make it more grammatically sound? E.g., "What's the weather forecast for today?" (~_^)

If you say "what is the weather", it sounds a little like you are asking for a definition or explanation of the concept of "weather". The other phrases are better, personally I usually say "How is the weather".