The story of how Walter Matthau came to be identified as Walter Matuschanskayasky is pretty well documented. He made it up himself. After arriving in the U.S. from Poland, his father had changed his name from Matuschansky to Matthow, which was Matthau's surname when he was born in 1920. Making fun of the original foreign-sounding name, Matthau would jokingly say it was Matuschanskayasky and some took him seriously.
This info is from his biography, Matthau, a Life, and I cannot shed any more light on these names. I cannot find the family in the census in any year with any name. His mother died with the name Matthow and there are possible sightings of his father in Ellis Island and draft records but nothing that gives a positive ID.
Goldie's story is quite different. I have no idea who made up the tale that her birth name was Studlendgehawn or what it is based on.
Here's her dad in 1920. He's listed as Rutledge, son of O.D. and Claire Hawn. Here he is again in 1910, this time listed as Edward R., son of Otto D. and Clara. All the way back through the census records for generations -- from Otto to Aaron to Alfred to Sampson to Christian to John -- all the way back to the first U.S. census -- the name is Hawn or Hahn or Haun.
The name is never anything at all like Studlendgehawn. As a matter of fact, I can't find a single person who ever existed with that name, can you?