Is zakat obligatory on money that is a trust (amanah) from the the mosque, and is it permissible to sometimes lend out the mosque's trust?

This money, donated to the mosque, is a trust (amanah (definition)) and there is no zakat (definition) on it because it does not have a specific owner. You are not permitted to lend this money, because it is a trust in your hands, and you may not dispose of it except for the purpose for which it was endowed. Furthermore, a condition for a lender is that he be someone whose donation is valid, and the supervisor of the endowment is not in that position.