What is the ruling on someone who prayed three rak'ahs in Tarawih, then stood up for a fourth, and performed the prostration of forgetfulness (sujud al-sahw)? And is there any opinion that allows for adding a fourth rak'ah?

What the person did is correct according to the Malikis. Whoever stands for a third rak'ah inadvertently in a supererogatory prayer and establishes it, should add a fourth rak'ah to it, except in certain supererogatory prayers. According to the Hanbalis, it is permissible to complete a daytime supererogatory prayer as four rak'ahs if one stood for the third inadvertently, whereas this is not permissible in a nighttime supererogatory prayer.