1. No way. Those were just some examples Binky imagined that were basically an exaggerated version of his allergy. A chef stepping on a peanut wouldn't have any effect...he isn't making lasagna with his shoe. For the middle name one, Binky is not allergic to the word, only the food. And once again for the last one, he is not allergic to the word "peanut" or the idea of one.
2. No, it's really doubtful. The store clerk would have had no reason to even care enough to listen to his conversation. And even if he did hear them, what would the clerk care? The clerk isn't a health specialist or something. For all he knew, Binky was buying the food for a friend or something.
3. No idea.