An old binder under a rash vest is exactly how I bind when swimming. It should be fine, just try to keep the binder as clean as possible when you get out so it lasts longer. But of course if it's old, its life is fading anyways.
As for the packing under board shorts, you could do your idea, or you could also try getting a jockstrap. If you can find one that has a pocket in it, that's even better. I don't know a lot about jocks other than that they're good for packing with, so hopefully someone else can chime in with specific suggestions. As for getting a cheap speedo or briefs, I don't know if you mean brief swimwear or brief underwear, but I'd highly recommend staying with swimwear because it will dry better when you get out of the water.