Is what befell my mother considered an affliction and expiation for sins due to her patience with the dog that causes her fear, and is she sinful if she stops feeding cats and birds until the dogs leave?

There is no blame upon your mother for not putting out food for the cats. A woman was tortured because of a cat she imprisoned; she neither fed it nor let it eat from the vermin of the earth. Feeding a cat is only obligatory for someone who has imprisoned it; rather, the duty is to set it free to seek its own food. If putting out food causes dogs to be drawn near and harm neighbors, then the rights of the neighbors take precedence over feeding cats. As for killing dogs, there is a detailed explanation: some scholars only permit the killing of rabid dogs, while others permit the killing of all dogs except those exempted for hunting or livestock. Some also consider it recommended to kill a dog that frightens and terrifies people, even if it is not aggressive. However, the black dog with two spots must be killed.