so I was wondering if I could run the processing program simultaneously with the arduino?
Arduino sketches run on an Arduino and don't rely on the computer to run.
Processing programs run on a computer and don't rely on the Arduino to run.
Unless you are starving for electricity, I don't see why Arduino can't do its thing while Processing does its.
Is it possible to run two arduino programs at once, though?
Sure you just need two Arduinos. The more cost effective option, though, is to combine the two sketches using multitasking techniques, such as those showcased in the Blink with Delay example.
Or any other way to play chords?
On one what I am assuming is a small piezo speaker?