Hi Vincent,
1) "John fell off the bicycle" is okay in all cases, but you may want to say "his bicycle," and then start the next sentence with "It" istead of repeating "His bicycle."
2) Don't say both "damaged" AND "broken." They mean the same thing. Choose one.
3) Did he fall BECAUSE the bike was damaged? Or was the bike become damaged as a result of the fall?
(a) John fell off his bicycle. It was damaged.
(b) John fell off his bicycle. It was [damaged] or [broken].
(c) John fell off his bicycle. It was damaged. If you want to use another word that means something other than "damaged" or "broken" you do not need to repeat the verb "was."
(d) John fell off his bicycle. It was not function /moved well. It was no longer functioning. It could no longer function. It did not function. It didn't steer well. It hadn't been steering well.
(e) John fell off his bicycle. It did not / could not move. It was badly / seriously damaged. Although you CAN have "badly" where you had it, it's more natural to put it before the adjective. Either badly or seriously is okay.