You are right about rats, thefox. Hadn't thought of them, but they will do that, too.
If you are not leaving eggs in the coop overnight, then lack of egg predation doesn't rule out opossum. Rat and opossum would be highest on my list.
Weasels and their relatives (such as mink and fisher) might bite a chicken on the rear, but this pattern of taking one every few nights, and not eating much of any one chicken would be very unusual for any of them. They are intelligent predators who often kill many at one time, more than they can possibly eat. This is very adaptive in the wild, where they can cache the excess kill and save it for future consumption. People think of this as selfish and vicious, but it is much the same as a human buying lots of steak at the grocery store and freezing it for future use. Anyway, this caching behavior is so characteristic of members of the weasel family, and the pattern you are seeing makes them really unlikely.
Also think about how large an animal could sneak into your coop. Is there any opening large enough for an opossum?
Please let us know if you solve them mystery!