One quick thought on ccarpet padding behind the door - moisture. If you opt to do this, I suggest sandwich'ing the padding between the existing moistore barrier and another one top. The second could be the home grown version with some clear heavy duty plastic with trusty duct tape would do the job.
The sound deadening material is expensive, but that would be my suggestion. Getting the door panels back on is hard enough, the thickness from the carpet padding would be make it impossible on my '95 E420 - the clearance on my panels is zippo!