What is the ruling on using usurious money that has been disposed of and paid by some brothers to the mosque, for expanding the mosque as a garage, or relocating the mosque's restrooms, or paying water, electricity, and gas bills? And what are the specific channels for spending these funds?

It is obligatory for whoever acquires unlawful money to spend it on the welfare of Muslims, such as hospitals and schools, and everything that benefits them generally. The more probable opinion is that it is permissible to build mosques with such money, as they are for the welfare of Muslims, as held by Imam An-Nawawi and Al-Ghazali. Therefore, there is no impediment to using such funds for the benefit of the mosque.