Is the prayer of one who, having erred in the third rak'ah, began reciting the tashahhud with the intention of the first tashahhud, then completed it, rose for the last rak'ah, and then said the salam, valid, or must it be repeated?

You were obligated to stand up for the fourth (rak'ah) immediately upon realizing that you had sat down in an improper place, and then perform the prostrations of forgetfulness (sujud al-sahw) afterward. Your continuation with the Tashahhud after realizing that the sitting was an addition is something that invalidates your prayer, and you must repeat it if you were aware of the ruling. However, if you were ignorant or had an interpretation, thinking that you must complete the Tashahhud once you had started it, then the correct view is that your prayer is valid, because the addition of an ignorant person is like the addition of someone who forgets.