Sorry, but I don't see the point here, maybe I am fatally blind. As I can see on your df output, your root-partition has 10GB.
According to the du output from post #28 this space is completely used by your files. There is nothing wrong with your system. I only ask me why you have a 6GB tarball in your /-directory, and you wonder why you have no space in your root-partition? Move it to your home-partition and you should be fine.