The S7 to Solingen will still be operating in June. You might wish to check on
www.bahn.de.
Either take the S7 at 18.03 from the Terminal Station and change at Solingen or (more realistic) take the S7 at 18.43 and change at Düsseldorf Main to a Regional Express Train from Platform 7. Luggage on the S7 should not be a problem. That train is never overcrowded when it starts at the airport. The RE train during rush-hour. Well....
You need two VRR tickets from the machine on the Platform for a C-Fare, 8,70 for an adult, 1,20 for a kid over the age of 6. Travel time is about an hour. Taxi should not be faster and will cost about 150 Euro for 70 km.