Will the one who builds a mosque within the boundaries of the Meccan Sacred Precinct receive the reward of multiplied prayers for every worshiper in it, or is his reward limited to building a house for him in Paradise, and the reward of multiplication is specific to the worshiper?

There is no reported information indicating that whoever builds a mosque receives a reward equivalent to that of someone who prays in it. Rather, the reward for the builder of a mosque is that Allah will build a house for him in Paradise. As for the muezzin, he is forgiven as far as his voice reaches, and he receives a reward equivalent to that of everyone who prays with him. The principle "The one who points to good is like the one who does it" testifies to this. It is not unlikely that this favor would also be attained by someone who appoints a muezzin for the mosque and assists him.