Re: Joker falling off a building
The falling motif was probably established in the comics.
From the Wikipedia article about The Joker:
In 1951, [Bill] Finger wrote an origin story for the Joker in Detective Comics #168 which introduced the concept of him formerly being the criminal Red Hood, and his disfigurement being the result of falling into a chemical vat.[18
That comic has been reprinted a few times in collections, so you can probably find it somewhere.
Dr. Howard, Dr. Fine, Dr. Howard: For duty and humanity!
--Men in Black (1934)