Tybalt is Juliet's cousin, originally, Tybalt wanted to duel Romeo instead. But since he and Juliet are married, he cannot fight Tybalt as he is basically his cousin-in-law, thus, Romeo rejects Tybalt's challenge. Mercutio finds this quite cowardly as a result he himself challenges Tybalt to fight.
Answer: True
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I believe that B. would be inappropriate in formal writing because its just kind of rude to say. I would re-word it as: "He dislikes being late" or something like that because "hate" is just a cruel and informal word to use.
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Yes you're answer (b) is correct. hope I helped. :)