Renting a car in Europe has its ups and downs. If you absolutely need to drive on the autobahn just to say you did it, no one will probably be able to talk you out of it. However, the heavily populated state of Nordrhein-Westphalen will teach you a new word with regards to the autobahn: Stau! The region you are visiting is heavily industrialized and also is (if memory serves) the most populated state in Germany. That means there are lots of other native drivers competing with you for those roads! I'd suggest sticking with the public transport, as that region has frequent and good train service. Driving in a busy German city is not much fun if you are not familiar with the traffic signs and traffic patterns, not to mention the numerous one way streets and streetcar tracks, etc.

As for authentic German, I'm not sure if Düsseldorf is the best city to satisfy stereotypes. The regional altbier is unique, but it is only one of many, many styles of German beer. Luckily, you are only a 30 minute train ride to Cologne, which is in my opinion a much more interesting town.