Just a thought but where was the air directed when you last shut the heat/cold controls off? Air will still come out of the vents when you are driving, even with everything off. If the system was last blowing on your feet before turning it off and the flap that controls the hot/cold air isn't moving and is stuck on hot, that could be the reason for the hot air.
Another question, if you turn on the AC, does the AC work or does it still blow hot air? Can you go from cold to hot? Can you change the fan to blow out of the vents, floor, and defrost, or does it only blow out by your feet? This will help us determine if your system is working properly.
Other than that, the floors/tunnel of the C5 get warm when driving as that is where the exhaust is routed. If you want to keep the interior colder, you can install kits to add insulation.