One big problem is that we only have one decay mode to work with: into two photons. Pretty much everything proposed expects that the particle (if such it is) will decay much more frequently into other things--but the backgrounds are gigantic. Decay into two photons is rare, but much cleaner. The other "decay modes" will take quite a bit more analysis to tease out, though some modes should be understood by the end of the year.
It is unexpected. It may be nothing very out of the ordinary, but everybody hopes not.
Is dark matter found in ultra-heavy particles, with masses comparable to a human cell nucleus? Answer: they don't know, and we don't have any handy way of testing that.
They suggest some indirect methods based on the Cosmic Microwave Background and what the Universe was doing at a very early era that we don't understand very well.
It looks like the cosmic rays we get on Earth come from quite a wide variety of sources. Most are protons, but some are heavier nuclei. The spectrum of energies shows several kinks in slope, as though one source pooped out and a fainter one took over. Currently we think quite a few come from a nearby supernova--which, among other things, would imply that the energy spectrum was different 20 million years ago.