It is a delightful story. I speak French fluently, and there are definitely some issues in the French in the story. Some of them come from transliteration errors inevitable with software translation programs, where the translated meaning is incorrect in the circumstances. Another issue, and the main one for me, is that no teenager would use the formal "vous" form of address with another teenager, especially one they want to become friends with. They would use the informal "tu" form. And there are other areas where idiomatic French would differ and not have a direct English counterpart.
But it doesn't detract from the story, even with the realization that it is improbable that a French teenager in Paris wouldn't know some English since English is routinely taught in high school and kids pick up on a lot of American culture (movies, music, and TV), along with the improbability that an American kid spending months in Paris each summer would not have picked up many bits and pieces of French despite himself. The premise works within the reality of the story.
Should anyone ever desire to include French within one of their stories, I would be happy to offer consulting services on how to say things appropriately.
R