Was what the Imam and the congregants did correct in the incident of the Imam's forgetfulness in the second rak'ah of the Asr prayer, where he did not audibly say the takbir for bowing, and the varied actions of the congregants in reminding him and bowing after him, and the Imam's prostration of forgetfulness at the end of the prayer?

What the Imam did, and whoever bowed then rose, is correct. And whoever delayed until the Imam said the takbir for prostration and rose, if he knew that the Imam had risen, then he is mistaken. If he did not know, there is no blame upon him. The prayer of all is valid. The prostration of forgetfulness for one who omits saying the takbir aloud for transitions is a matter of dispute among scholars: For the Hanbalis, there are two narrations: one states it is not prescribed, and the other states it is prescribed and is recommended. The Shafi'is do not see prostration for this at all. The Malikis do not see prostration except if one omits uttering two or more takbirs other than the takbirat al-ihram or the Eid takbir.