You have mentioned, "Suppose we do have nine spatial dimensions and some of those dimensions are curled up. If you're working at distances that are much bigger than the curled up dimensions then the law looks like $1/r^2$. And when you're working at distances that are much smaller than the curled up dimensions the law looks like $1/r^8$."
You are only supposing to have nine spatial dimensions. It would have been better if you would have clarified how string theory tells us to suppose "nine" - not "twelve".