Most replacement exhausts from the dealership or even the aftermarket will have a new pipe included with the muffler you'll need to purchase. Get the whole thing. It will be a lot easier to take care of if you do.
As far as claiming money because of speed bumps? No. It looks as though the muffler and pipe (and surrounding areas) are rusted. This is just normal wear/tear on your car. You'll need to replace the parts. Nobody is going to pay for it and there's no way you could prove something like a speed bump has caused the issue in the first place.