Couple of suggestions, first regarding the headless pigeon; sparrowhawks are not that good at catching prey of pigeon size, a bit big to be their normal prey, they certainly wouldn't have been able to carry the carcass away with them so it may have been an opportunistic sparrow hawk kill with the hawk being frightened off from the carcass before it tucked in. Birds of prey leave characteristic marks on the stalks of the feathers of their kills as they pluck them out to get at the meat it often mangles and bend and sometimes punctures the stem of the feather. there will also be conspicuous puncture wounds from where the talons puncture the carcass, buzzards and red kites aren't good enough hunters to take a pigeon unless it's a squab or injured to begin with, they tend to eat a lot of worms, beetles, carrion and some small rodents.
Regarding the rabbit remains, badger would be a good culprit I'd have thought but it must have been startled off it's meal to have left the hind quarters behind, badgers often leave neat little piles of innards behind, I think someone already mentioned hedgehogs which they will eat from underneath leaving a shell of spines a pile of intestines and sometimes feet. I have a couple of times found sign like this which was definitely fox and was caused by a fox pulling a rabbit out of a snare leaving part of the animal in the snare and taking the rest with it. Smaller mustelids (stoats and weasels for example) sometimes disembowel their prey but aren't big enough to be able to tear a rabbit in half like that.