I'd say that brave and courageous have the same heroic meaning. Bold can mean the same but you're right, it's more likely to have a connotation of someone who is rudely pushing forward, someone who has no problem speaking to a person of authority or something like that. It's probably more a question of confidence whereas being brave or courageous is often used to describe a person who will act in spite of fear.

Brave and courageous are near perfect synonyms. Bold can have the connotation "brave" but can also mean "very confident" and (as you suggest) "impudent." It can have a negative connotation, but it need not be negative. It depends on the context. A rather famous line from the opening of an iconic 1960s US TV show (Star Trek) goes something like "To boldy go where no man has gone before" -- and in that context it was supposed to be (quite) positive.

I agree that bold doesn't necessarily have a bad connotation. A bold color (which would be like a very vibrant color). Someone with gumption, sass, or "balls" as we might say which certainly could get you into trouble, but I think bold just assumes that someone is not afraid to "go for it,"-- to take big action. but yeah, there's a slightly different connotation than bravery/courage which imply a hero whereas bold could be applied to someone who is a hero or a villain, or just daring. I really like the word.

I think the three mean the same, and bold doesn't have a negative meaning.