What is the ruling on one who doubts counting a rak'ah in which he caught the Imam bowing, and is unsure whether the Imam said "Allahu Akbar" after rising from rukoo' or during it?

Your prayer is valid, as you counted the ruk'ah, and you caught the Imam bowing, then he stood up a short while after you bowed. As for someone who finds the Imam bowing and doubts whether they caught the ruk'ah, they do not count it and should disregard it, as Muhammad 'Alish al-Maliki mentioned in "Minah al-Jalil." We advise you to ignore obsessive thoughts and disregard them.