Just a short note:
Deutsche Bahn has decided, that since 12th december only a few High Speed trains will stop at Düsseldorf Fernbahnhof.
There is a desk in the Train Station, reached by the Skytrain from the main terminal, where you can buy tickets. If you are able to catch an IC or ICE you are able to buy tickets on board the train (surcharge), but not oboard regionla trains.
There is no need to take a high Speed into Cologe, as the regional train is only a few minutes slower. The directions of the trains to Cologne are either to Aachen or Koblenz.
And yes, it is worth saving the money.