[uncle_bob]

I am wondering if the 7809 will help handling/curbing this. Would it just mimic this oscillation in the 9v output or will it somehow attenuate the gap? What about the Arduino? Can this mangle the reference voltage?

The Voltage regulator does does just that. It takes any input voltage within it's operating parameters (something like 11 to 30v) and gives a steady output regardless of it's input.

They do get hotter the higher the voltage they have to drop.