It sounds tight-ish but doable. Munich airport is usually spoken of as extremely efficient for connections -- the minimum connection time for Lufthansa international-domestic is just 30 minutes. They consider this swift connection a definite selling point, and they go to quite some effort to keep it, including such things as proactively fast-tracking late ...
You do need an emission sticker for the inner parts of both cities. Here are links to maps showing the zone for Düsseldorf and Cologne (Köln) to see if your destination is part of the zone. If these links become invalid, just use Google's picture search for "Umweltzone [name of city]" and you should get decent results.
First I'd ask the car rental company ...
On www.gratisparken.de you can find a lot of free parking places in and around Düsseldorf. The closest to the airport are on these streets:
And on www.duesseldorf.de there is also a map with all official P&R in Düsseldorf.