Does my promise to my mother to donate to a specific mosque constitute a debt upon me after her death, and am I obligated to fulfill it to the same mosque, or is it permissible for it to be to another, or is this promise absolved from my responsibility?

You must fulfill your promise to your mother by donating money to the mosque she desires, as this is an act of filial piety and kindness towards her. If the aforementioned mosque no longer needs the money, then donate it to another mosque. The most important thing is to fulfill the promise as an act of devotion to your mother.