Is my prayer valid, given that I caught the first Tashahhud with the Imam in the Dhuhr prayer, then after the Tashahhud, he prayed three rak'ahs by mistake, so I completed the remaining two rak'ahs after his Tasleem, and he performed the prostration of forgetfulness (sujud al-sahw) after the worshippers alerted him, and I performed the prostration of forgetfulness after completing the two rak'ahs?

If you prayed a fifth rak'ah with the Imam, not certain that it was a fifth, then performed two rak'ahs after the Imam's tasleem, your prayer is valid. Scholars have differed regarding counting the extra rak'ah; some preferred counting it, while others did not. As for the sujood al-sahw (prostration of forgetfulness) for the Imam, it is performed after the tasleem if he added a rak'ah. The latecomer ma'mum (follower) does not prostrate with the Imam if the Imam's sujood is after the tasleem; rather, he prostrates after completing his own prayer, unless the forgetfulness occurred before he joined the Imam.