If you are sure that the Masjid will remain under the management of reasonable persons and will not be influenced by extreme Wahabi groups, then you can help them.
It is better to help poor Mo'mineen in poor areas or poor countries where they don't have a Masjid.
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.
This matter has different opinions among the Maraaji' of Taqleed who are the highest in Islamic knowledge. Many of them e.g. Ayatullah Sistani and Ayatullah Sayyed Sa'eed al-Hakeem say that non Muslims are not allowed to enter Masjid according to obligatory precaution but they are allowed to inter all other religious places like Islamic centers and Husainiyya and Imam bargah etc.
'Other Scholars e.g. Ayatullah Kho'ee and Ayatullah Waheed Khurasani say that non Muslims are allowed to enter usual Masjids "except Masjid al-Haraam " if they don't carry with them any item which can make the Masjid Najis.
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