After opening up the walls, Tig fashioned a remote camera out of some wire and duct tape. With the camera firmly attached to a wand of metal, he fished the camera into the hole and took pictures. Actually, it was a little more complicated than that. The camera was wired to the laptop, so that we could see the images from the camera. It was much like microsurgery, with Tig manipulating the camera, while I operate the laptop and snapped pictures using remote commands.

Here were some of our suspects:

pvc on wood
We think the vertical drain pipe is pushing against this wood header, that supports the wall studs.
pipe on pvc

We also think that the copper pipe and pvc drain pipe from the sink are too close.

Confirmed by the next shot.

pipe on pvc2

The copper pipe, then rubs against the wall studs (below)
pipe on wall stud

So our theory is that the pvc pipe expands from the heat of the hot water draining, pushes against the copper pipe, which is right next to it. The copper pipe is already backed up against several wall studs. The pressure causes everything to rub against each other, which produces the creaking/cracking sounds.