What is the ruling on accusing one's brother of being a cuckold (dayooth), and does it necessitate ta'zir (chastisement)? And does the sin cease with repentance and seeking forgiveness, or is it obligatory to seek the brother's forgiveness or apply ta'zir? And how is ta'zir carried out in this situation?

Whoever utters the word "ya dayyuth" (O pimp/cuckold) deserves ta'zir (a discretionary punishment), and a dayyuth is one who allows men access to his wife. Ta'zir is carried out by the Imam, not by individuals. You are not required to demand the court inflict ta'zir upon you, and it is preferable for you to conceal your situation. You must seek forgiveness from your brother.