Just adding... "To kid" usually requires someone to kid, whereas "to joke" doesn't need that. So "are you kidding me?" and "he's kidding us" are possible, but not "are you joking me" or "he's joking us" - you would use "joking with (someone)" if you had to indicate who was receiving the teasing. I suppose this may be because "joke" can function as a noun, as Steven2 pointed out. But you can also write "(I'm/we're/she's) just kidding." if there is no deliberate target for the joke.

They mean the same thing. "joke" can also be used as a noun meaning something funny. eg. "That joke was really funny". You cannot use kid in that sentence.

They mean the same thing. Afaik, "joke" is the more formal usage though it is also used in casual convesation. "Kid" is used more casually. Positive: "I was just joking". "I was just kidding you" Negative (not): "I kid you not" or "I am not joking"