I agree with the others, I don't see the point of renting a car for that itenerary. For me, it'd be more of a hassle than a convenience. Don't underestimate the ease and convenience of trains in Denmark and Germany - and the inconvenience of bringing a car to an old town or city.
Copenhagen, Aarhus, Odense and Roskilde (and Lübeck for that matter) are all connected by very regular trains - and Billund can be reached with a little bus ride from Vejle.
I'd recommend taking a look at www.journeyplanner.dk for public transport timetables in Denmark - and get your train tickets on dsb.dk/en (for international tickets: compare with bahn.com).
However, those are nice towns and cities, and I think you can have a great and fun holiday in these places.