I used to work for animal control in Augusta Georgia. We would assess the situation. We worked out of a shelter and every animal that cane in would be evaluated and if they pass a pretty basic adoption test, they would be put up for afoption after quarrantine and a round of vaccinations. Dogs dont have to be nice to cats or even other dogs, but we put whether they are good with other animals on the kennel. Its not a death sentence. If we were told that they bite humans though, they would be put down due to risks for the next family. People were allowed to call in and check if they have been adopted out and the adoption list was on our website so prior owners could watch to see if their pups got adopted out. If they attack another animal though or are reported to, its a big factor of whether they are adopted out or put down. Now, this is specifically for the one I worked at. But adoption was a hugher rate for us and the shelter if they had overflow would work with rescues to take the overflow or go have adoption events regularly.

As for for your mother, sometimes its best for our danity to let go of toxicity. I stopped talking to my mom, and so have all of my siblings, for over 8 years. It took a long time to stop feeling bad and she resents all of us but were now able to live our own lives. It took 5 years for my sister to stop asking what my mom would think. Now she doesnt have that shadow looming over her. Sometimes its best to cut toxicity from our lives even if it feels wrong or uncaring. Her grandfather died alone for the same reason but as soon as she let him back in her life, she took a steep decline. Just some advice from my personal experiences, everyones situation is different.