Could I run a conventional 2 stroke engine on an ethanol and biodiesel mixture?

A two stroke diesel probably could be run on biodiesel. But if you are asking about a two stroke gasoline engine operating on an ethanol and biodiesel mixture, the answer would be no. Biodiesel is not a substitute for gasoline and I have no idea what would happen if you added ethanol to it. My guess is you'll ruin the engine.