Is zakat obligatory on a rented laundry shop, the purchase of which was financed three years ago by a bank loan, and is it permissible to spend the zakat money on building or furnishing a mosque?

Zakat is not obligatory on the laundromat itself or its equipment. Rather, zakat becomes obligatory on its income if it reaches the nisab (minimum threshold) and a hawl (full lunar year) passes over it. Zakat is not to be given for the construction of mosques.