The house in which Miss Cooper lives stood on the back lot of Republic. For multiple decades, It has appeared frequently in movies and on TV in quite an array of settings and atmospheres. It was used for the Smith family residence in Meet Me in St. Louis (1944)
In this episode, there is an extension phone upstairs outside the boys' bedroom in the Mapleton Drive home. In some episodes, Wally and Beaver must come downstairs to answer a call.
In cautioning Beaver about the possibility that Miss Cooper may be a witch, Larry resorts to explaining the horrors experienced by two other kids in the fairy tale "Hansel and Gretchen." Larry confuses the title for "Hanzel and Gretel," the German fairy tale popularly known as one of the stories collected by the Brothers Grimm, i.e. Jacob Grimm and Wilhelm Grimm, and published with other such stories as "Grimm's Fairy Tales" in 1812.
This episode gives another clue as to Mayfield's location. If Ward's father knew the Cooper house was the first to get electricity, it's likely Mayfield is near Shaker Heights, Ohio, where Ward grew up.