Is my prayer valid if I did not hear the Imam's takbir for rising from rukoo' (bowing), so I bowed, rose, and completed the prayer with him?

If you are able to perform the bowing (rukūʿ) that the Imam preceded you in, then your prayer is valid. Ibn Qudamah mentioned in "Al-Mughni" that if a congregant (ma’mūm) is preceded by a complete pillar due to an excuse such as drowsiness or crowding, then he should perform what he was preceded in and catch up with his Imam, and there is nothing upon him.