Is it permissible for me to go out and pray in the mosque's courtyard, taking the necessary precautions, or should I pray at home, especially since my brothers do not pray in congregation with me?

A Muslim must obey his father by praying at home as long as mosques are closed. He is not religiously obligated to pray at the mosque's threshold. He will be rewarded — Allah willing — with the reward of congregational prayer, as long as he was keen on it and would not have missed it were it not for the closure.