Must I perform all the obligatory prayers in the mosque, or is it sufficient for me to perform only one prayer daily in obedience to my father's command, especially given the current circumstances that might affect my studies?

If you fear harm from going to the mosque, then you must obey your father in not going, and you can pray in congregation with him at home or with your family members. If there is no harm and your father does not object, then go to the mosque to attain the virtue of praying therein.