Here in Australia the earlier iconic Holden "panel van", loved by surfies and tradies, used to have a large window which hinged up from the roof at the back. I'm sure this was intended only for use when parked but it was common to see people driving around with it open on a hot day.
Of course being a van with only front seats it was much more practical for amorous activities as well. Surfies would often have a mattress in the back!
It looks like Holden might be planning to make them again
http://www.holden.com.au/concept-cars/sandman-panel-van-
Back to John's problem. The cheapest and quickest solution to the exhaust stains has to be aftermarket exhaust tips that point downwards. But I suspect there's also a leak into the cabin from either the front or rear of the car.