Is the prayer of the Imam valid if he bowed without saying "Allahu Akbar" or if the congregants did not hear his "takbir," then he stood up from the bowing position to say "takbir" and bowed again, and after the prayer, performed the prostration of forgetfulness (sujud al-sahw)?

What the Imam did by returning to the standing position after settling into rukoo' (bowing) is a clear mistake. The obligation upon him was to continue in his rukoo'. The takbeer (saying "Allahu Akbar") for transition is a Sunnah according to the majority of scholars, and obligatory according to the Hanbalis, and pronouncing it loudly is a Sunnah. Whoever leaves the takbeer due to forgetfulness or ignorance should not return to it, and it is compensated for by the prostration of forgetfulness (sujood as-sahw). Whoever returns intentionally, knowing the ruling, his prayer is invalidated. And whoever returns out of ignorance, his prayer is valid. The prayer of the congregants is valid, except for anyone who knew that the Imam intentionally violated the ruling, in which case they must repeat it. The Hanbali madhhab holds that they all must repeat it.