Is the vowed amount required to be paid after the condition is met? Is it permissible to place it in the mosque? And is there a sin incurred for delaying its payment?

A vow is not binding unless it is made with a صيغة (formulation) that indicates commitment. A vow contingent on the will of Allah does not necessitate fulfillment according to the majority of scholars, but it is preferable to fulfill what was promised. In the case of fulfilling the vow, the amount must be donated to the intended recipient. If no specific recipient was intended, then it is sufficient to give it to the poor, or to a mosque, or for anything in the path of Allah. Making vows is disliked, as there is evidence of its prohibition.