Engaging in haram acts does not mean one abandons their devotion to Allah ta'ala and their 'ibadah.
Indeed, it is even more important to stress on that, for the purpose of hopefully empowering oneself to become more obedient to God and stay away from sin.
Yes, this man should continue to pray, fast, and perform everything else required by him as a Muslim. If done in the correct way, they will also be valid as well.
Yes, we are allowed to pray at home especially when the Mosque is closed. Friday prayer has its conditions where ever it was established e.g. minimum number of participant is five, and the Imam of Jum'ah must be 'Adil (pious believer who does not commit sins) and other conditions.
According to the view of our esteemed jurists, attending Salat al-Jumu'ah (Friday Prayer) is not wajib, but only Mustahab, and you are able to either attend, or pray your Dhuhr and 'Asr at home. You have this option to choose.
Of course, it is always recommended to pray in the form of jama'ah, or congregation, so it is encouraged to try to pray your wajib prayers in jama'ah in any opportunity you have.
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