Am I religiously sinful after the death of a small cat due to a door falling on it, especially since I saw how the door was placed and did not intervene, and I heard it fall but did not rush to check on it? Is this considered intentional killing on my part, or does the sin fall upon my mother and brother?

There is no sin upon one who unintentionally causes the death of a cat, because intent is a condition for accountability, and a person is not held accountable for mistakes, due to Allah's saying: "And there is no blame upon you for that in which you have erred, but [only for] what your hearts have intended. And ever is Allah Forgiving and Merciful," and due to the Prophet's (peace be upon him) saying: "Indeed, Allah has pardoned for my Ummah their mistakes, their forgetfulness, and what they are coerced into."