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.