Do they have a stretch waistband?

Yes they have a stretch waistband.

Would the Caviar pass as Black?

I just bought the color Caviar and it's jet black.

Which color is closer to khaki than brown, "Morel" or "Bungee cord"?

I ordered the Morel color hoping it was "closer" to a darker khaki and while I like the color I would not consider what I received as being close to khaki at all (or close to the image showing the color I ordered for that matter, definitely more brown than expected). If I had to compare the color I received to common browns I would put it between the color of work boot leather and hot chocolate with a little milk. something like this.. Khaki > > > leather (common brown work boots) > morel > hot chocolate. I realize that is a stretch of a description, but hopefully it helps. For what it is worth I have not seen these in bungee cord, so I still can't really say which of the two (morel vs bungee) is closer to khaki... one additional note; based on the differences in the designs I have received under this same description, I believe there could be some variation in what is being shipped as I have ordered the same model/style of these pants 3 times and while I ordered different colors there were some substantial differences in the design of the pants even though the only difference was supposed to be color. This isn't a problem for me, I like the pants for the price, but I wouldn't be surprised if the colors were as variable as the style.