It has been sometime, 10 years now, that I rode from Den Haag, Utrecht, Arnhem, Venlo, and to Maastricht. I was there just as the tulips popped and it was awesome. You'll find that Arnhem to Maastricht will dispel the idea The Netherlands does not have hills. I have also done a bike/barge trip through some of the same area and enjoyed that as well. I like Arnhem.
I have also done the the Rhine from Düsseldorf to Koblenz to Trier and that is a very nice trip. I laid over in Cochem and enjoyed it. But that is the reverse direction of your trip.
The last time I went to Europe I flew British Air from Phoenix to London Heathrow non stop and used the CTC Bike Bag http://www.ctcshop.org.uk/cycling/p/CTC/22/CTC_Plastic_Bike_Bag/5360041697/
It worked great especially when I broke it down with the front wheel tapped to the frame center. Made a nice compact package to move around and handle. Seemed to had been handled properly too as there were no issues after flying both ways packed that way.