Is the prostration of forgetfulness (sujud al-sahw) obligatory for someone who joins the imam, thinking he is in the tashahhud or a specific rak'ah, only to find him in a different part of the prayer? And is the prostration of forgetfulness obligatory for someone who joins the imam late, then forgets (makes a mistake) in the takbeers other than the opening takbeer, or forgets in the rak'ahs he completes after the imam's salutation?

When you come to prayer, do as the Imam does, then make up for what you missed. It is not necessary to know which rak'ah the Imam has reached. If a worshipper makes a mistake (sahu) behind the Imam, there is no prostration of forgetfulness (sajdat al-sahu) required of him, according to the majority of scholars, because the Imam bears this forgetfulness on his behalf. The worshipper prostrates with the Imam if the Imam makes a mistake. However, if you are a latecomer and you make a mistake while making up for what you missed after the Imam has finished the prayer, then you must perform the prostration of forgetfulness.