I rescued the shelter’s oldest dog, fully aware that she had only a month left, and I was determined to fill her final days with joy.

When my new neighbor Claire moved in about six months ago, I expected her to be just another polite, minding-your-own-business person. In her 40s and living with her 16-year-old son Adam, she initially seemed pleasant. I even lent her our ladder, garden hose, and let her use our outdoor grill when we weren’t around—simple neighborly…