Ellington's childhood piano teacher was, indeed, Mrs. Marietta Clinkscales. See "The Duke Ellington Reader," p. 6. The line about "don't sit down at the piano after..." is just Ellington recycling an old joke, a la the one about the New York concert debut of some new piano virtuoso, say the young Horowitz. Rachmaninoff and Fritz Kreisler are seated in the front row, and after Horowitz rattles off a piece or two, Rachmaninoff says, "Is it getting hot in here?" To which Kreisler replies, "Not for violinists."
Accoding to Steve Voce it's something Duke often said when he was following on stage a piano player he admired. Voce mentions Nat Pierce as an example but I'd like to imagine Ellington saying it about Willie the Lion or James P Johnson -- of course in those cases it could be a true story.