Is there a sin upon my mother for having strangled a cat to death when she was young, believing that the cat was the one that ate the chicken eggs? And what must she do if there is a sin upon her?

Killing cats or other non-harmful animals is not permissible except for a legitimate Shar'i reason, such as if they are harmful, or carry dangerous diseases and cannot be eliminated except by killing. Whoever transgresses in killing them must repent and seek forgiveness. There is no specific expiation for killing cats, and a child who has not reached puberty bears no sin.