Hi Mike - I would suggest driving from Erfurt to Mannhelm or Frankfurt airport, (whichever is nearest) and flying to Vienna, then flying back, pick up the car and drive to Gelsenkirshen. - To drive from Erfurt to Vienna then back to Gelsenkirshen will be almost impossible unless you are super human and don't need any sleep!lol
Or you could look into the possibility of taking the train.

Hope this helps.