Is the prayer valid if I joined the Imam in the last rak'ah of the Dhuhr prayer, then stood up for the second rak'ah after the Imam had said the Taslim, and prayed two rak'ahs and sat for the Tashahhud, then stood up for the last rak'ah and sat for the Tashahhud?

You should have sat in what was for you the second rak'ah and recited the tashahhud, then stood for the third without sitting, and then for the fourth. Since you did not do that, you omitted one of the obligatory acts of prayer. Performing the tashahhud in the third rak'ah does not benefit you according to the majority of scholars, in contrast to Abu Hanifa's view. You should have performed the prostration of forgetfulness (sujood al-sahw) for the omission and for reciting the tashahhud in the wrong place. Since you did not perform the prostration of forgetfulness, it appears that you must repeat the prayer, because some scholars hold that if one omits the tashahhud and its sitting, and does not perform the prostration of forgetfulness, their prayer is invalid. However, according to Abu Hanifa's view, nothing is due from you, but the view of the majority is prioritized.