What is the ruling on the follower saying "Allahu Akbar" loudly in prayer, and is he required to make up previous prayers if saying "Allahu Akbar" was obligatory and he did not do so?

The concurrence of the Imam in the utterances of prayer – other than the Takbirat al-Ihram (definition) and the Taslim (definition) – is neither disliked nor does it invalidate the prayer. Your prayer is valid even if you say the Takbir simultaneously with the Imam. Doubt concerning an act of worship after its completion does not affect its validity. As for obsessive whispers (Waswas), they must be repelled and not indulged in.