What is the ruling on the action of the mother and what is incumbent upon her to clear her conscience after discovering a shortage in the amount of a trust deposited with her by her cousin, and is it obligatory to inform the cousin of this?

This dowry is a trust with your mother for your cousin. If she was negligent in preserving the trust, then she is liable. Otherwise, she is not liable, because a trustee is not liable. Your sister must return the amount she took to its owner, based on the hadith: "The hand is responsible for what it took until it returns it." It is not a condition to inform the owner when returning the amount.