reported speech

I would use "had fallen." Because the simple past verb "said" refers to an action that occurred in the past, it would make sense to use the past perfect "had fallen" to indicate that the occurence of his birthday predated John's statement.

it depends. certainly if you're quoting, then you must sitck with what you're given. (you can always insert brackets to indicate deviations from the original that you require to fit into your sentence structure.) barring that, if you're paraphrashing only then defer to had fallen because you need the past perfect to indicate that an event occurred in the past even thought you are in the present.

"Had fallen". Always 'step back' one from the tense/aspect used in the direct speech when reporting indirect speech.

English teacher is correct. I would only add that you are not required to repeat John's idiom if you aren't quoting him. In this case the idiom "fell on" is awkward. To say "my birthday fell on last Friday" makes it sound like a catastrophe. A better sentence would be "My birthday was last Friday." In paraphrasing, feel free to improve: John said that his birthday was last Friday. Whether you change "last" to "the previous" depends on which is clearer in context.