Is it permissible to perform Salat (prayer) and Jumu'ah (Friday prayer) in a mosque built with money, the majority of which is illicit (such as from drugs and gambling)? And is it permissible to pray at home instead, especially since those in charge of it love innovations and hate the Sunnah?

It is permissible to build mosques with ill-gotten money whose owner is unknown or to whom it cannot be returned. Prayer in these mosques is permissible without any dislike. As for praying behind an innovator imam, if no other imam is available, then it is permissible to pray behind him by consensus of the Ahl al-Sunnah (definition) wal-Jama'ah, especially for Friday and Eid prayers. If it is possible to pray behind someone else, that is better. However, congregational prayer behind a dissolute person is better than praying alone. The prayer is valid behind him according to Al-Shafi'i and Abu Hanifa, while there is disagreement in the Maliki and Hanbali madhhabs. A Muslim is not excused from praying in the mosque due to an innovation or an abomination; rather, he should attend and denounce the abomination to the best of his ability.