What is the ruling on the prayer of one who intended to complete the rak'ah with the Imam after the Imam's prostration of forgetfulness, and of one who intended to complete it individually?

It is permissible for a latecomer (masbuq) to rejoin his imam if he stood to make up what he missed and it became clear that the imam had missed a rak'ah. It is also permissible for him to leave his imam and complete the missing rak'ah alone. What your brother did is correct, and his prayer is valid. The prayer of the latecomers who left their imam and completed the missing rak'ah alone, and unintentionally omitted the prostration of forgetfulness (sujud al-sahw), is also valid.