I think you nailed it about the right arm thing. For the majority of people, the right arm is the dominant arm, so you would be paying a dearer price to give it up.
As for left nut, I have a couple theories. One is simple phonics; the "ft" sound adds an inherent level of humor to the phrase. The other theory is that the word "right" is ambiguous. Rather than clearly signifying a right/left dichotomy, it could be taken to mean right/wrong, which creates a momentary confusion that dulls the impact of what you're saying.
Another possibility is that the left, or "sinister", nut is the money nut, the nut that gets things done. I'm not a doctor, but I'm willing to pretend to be one if you need me to back that theory up.