The part they replaced is displayed here
http://parts.bmwofsouthatlanta.com/p...107842385.html
code 28107842385
The noises described in 2) and 3) are very different actually. You may be right that the noises described in 1) and 2) both come from the differential and not the DCT itself! My differential does seem to have a little slack. It makes sense that it could make such a noise when changing from no load or engine braking/reverse to normal load as it does in 1) and 2). And as you said, that would not be such a big concern. Its the DCT which is expensive to replace...
The more quiet clicking and whining noises are clearly related to the DCT preparing to change gears and the revs. And at the same time they are related to temperature as explained...
You do not get any of these noises at all?