Call or email the GT provider. They will take you from CGN to DUS. It should only take about 45 minutes. This will be the easiest way for you. I was going to use them last month to go from CGN to FRA, but my trip got revised. I used them to go from the hotel in CGN to the airport. This will be easier than trying to connect with trains and all that mess.
No experience with the lounge in DUS. If you go through AMS, there are showers in the KLM lounge there.

Good luck.