[AllanG]

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 (and consequently the readings)?

The output of the 7809 will remain constant even if the input voltage changes. If I remember correctly, the 7809 should handle input voltages from around 11 to 35 volts.

Hope this helps.