What is the ruling on someone who prayed Dhuhr and entered the third rak'ah, but the imam stood for a fifth rak'ah, so he followed him, then he realized that it was invalid, so he prayed Dhuhr again after the Asr Adhan? And what is the ruling on the prayer of someone who joined him in making up the pre-Dhuhr Sunnah prayer?

If the Imam stands for a fifth rak'ah, the worshiper should not follow him if he knows about the extra rak'ah, even if he is a latecomer to the prayer (masbuq). If the latecomer follows the Imam in the fifth rak'ah unknowingly, there is a difference of opinion regarding the validity of that rak'ah. Sheikh Ibn 'Uthaymeen, may Allah have mercy on him, was of the opinion that the rak'ah is valid if the worshiper was unaware that it was the fifth rak'ah for the Imam, and he is not required to perform another rak'ah. According to the Sheikh's preferred view, your brother's Dhuhr prayer is valid, though repeating it is better and clears his conscience more. Your following him in prayer is valid according to the preponderant opinion that it is permissible for someone praying an obligatory prayer to follow someone praying a supererogatory prayer.