Question: Because the right and left brain control not only different parts of the body but also feelings and speech (along with lots of other things), how would a left handed person's brain be different to a right handed person's brain?

As far as I know, it doesn’t have an impact on other processing things, like language. So language is processed in the left temporal lobe, and this is true for right handers, and the vast majority of left handers. I am fairly sure that this goes for other areas with a specific function too!