Edith the Talking Head is a take off on Edith Head, the legendary Hollywood costume designer (whose services this Edith would obviously not need, being just a head).
The bit starting at 6:16 is a parody of "Pinky the cat." Pinky became a viral sensation as the rescue cat a local sheriff attempted to make an adoption plea for, only to have the cat react wildly and bite the sheriff after climbing around his leg.