Is the congregant required to wait for the Imam to bow before bowing himself, or should he bow immediately after hearing the Imam's takbir, and what if he misses the bowing?

If you hear the Imam's takbir for bowing, you may begin bowing immediately. However, it is preferable to delay slightly until the Imam actually bows, though this is not obligatory. If you miss the bowing, meaning the Imam bows before you, then you should bow and join him, and your prayer will be valid.