I'm a U.S. native speaker and to me it sounds correct, idiomatic, and natural. It is ambiguous, though. "It is a little rainy today" usually means what it says. "It is a bit rainy today" could mean what it says (there is a _very_ small amount of rain today), or it could be humorous understatement. I can imagine a conversation like this: "You're drenched from top to bottom! It must be pouring out there." "Yes, it's a bit rainy today."

Yes. It's correct and natural.

Yes, that sentence sounds natural and is correct. You could also say "It's a little rainy today."