Unfortunately there are just some people you simply cannot educate.
55fingers- your poor family dog sounds like he may be diabetic now, or has some other health problem now that he's getting older, & that's truly horrific, to have stuck him outside like that in the cold. I believe I'd have called Animal Control too- again, you have my deepest sympathy for what you & your family's animals have suffered through from your clueless mother. I don't know if there's any way to legally bar her from owning any more pets, but in a perfect world, there sure would be. Maybe AC will have some ideas? I hope so. It really puts you in an awful spot.
(My mom had her issues but poor animal care was never one of them- she took great care of any pets she had- better than people, I'd have to say. )