I always find it exciting when common materials suddenly turn out to be awesome. While this certainly needs more research, we don't have a barrier of "the applications are endless, but it's gonna take a century or so until we can produce it economically".
(you actually want the heat to be conducted away from the CPU as fast as possible - which is why you fan it with air - to increase the heat transport - a thermal insulator on your board is pretty much not that useful)
Yes, so was I - if it can't conduct heat to somewhere else because it's heat conductivity is low, it obviously has to accumulate it locally, which mean temperature ramps up till it melts.
Partially from the finite resistance (imperfect electrical conductivity) of the material. Partially also from the need to 'forget' states in a processing device (otherwise Maxwell's demon would work) - that's dictated by the growth of entropy.
I suspect what you actually want as material is a high-temperature superconductor - zero resistance, and all heat carried away very quickly (i.e. high electrical and thermal conductivity).