Is praying Dhuhr on Friday at home, in congregation, two rak'ahs aloud and two rak'ahs silently, valid, or must it be four rak'ahs silently?

It is obligatory for those not required to attend Jumu'ah prayer, such as the sick and women, to pray Dhuhr. Dhuhr prayer should never be recited aloud, whether prayed on Friday or any other day. Reciting aloud or silently is a Sunnah according to the majority of scholars, and obligatory according to the Hanafis. As for your past prayers, they are valid, Allah willing, and do not need to be repeated. According to the Malikis and Hanbalis, it is recommended to perform the prostration of forgetfulness (sujud al-sahw) for one who recited aloud forgetfully. However, one who did so intentionally does not perform this prostration. The Shafi'is do not consider sujud al-sahw necessary for one who violates this Sunnah, citing the hadith of Abu Qatadah that the Prophet, peace and blessings be upon him, would sometimes let them hear a verse in the first two rak'ahs of Dhuhr prayer.