To round off a few things here:
a. the Hamburg-Aarhus/Copenhagen route is split between ICE-TDs and IC3s. As others have noted, the ICE-TDs are the better trains, and operate 4 x day to Copenhagen but only 2 x day to Aarhus. The latter services come as a pair from Berlin, then divide at Hamburg Hbf.
b. The DSB IC4s are still causing problems and currently only seem to be running on Aalborg-Aarhus-Esbjerg Regional trains, as an extended running-in trial.
c. The trains to Copenhagen use the train ferry. You have to leave the train during the voyage (which takes c. 45mins), ie. leave bags behind. The train uses the same deck as the trucks and for safety reasons everyone must leave that deck during the voyage. It makes a nice interlude in the middle of what is otherwise quite a long journey...
d. The ferry can take 2 IC3s joined together, just on one track, but only one ICE-TD (although in principle there are two tracks so two could be put on the ferry). In the good old days, LHCS was split up onto the ferry, ie. the first three or four carriages were shunted onto the ferry, were detached and then the rest of the train was shunted back off the ramp leading to the ferry and switched into the parallel track on the train deck. But that was 20 years ago...