1.) Parallel axis is typically used for larger scale structures. For solid and honeycomb laminates, we typically use classic lamination theory, which includes some very important coupling terms.
2.) The only place I have seen carbon honeycomb is in very high temperature applications. If I remember correctly, they start with a "standard" honeycomb and then place them in a high temperature, carbon rich oven. When the part comes out you are left with a carbon/carbon honeycomb. And yes, it is VERY brittle. I'll take a look around and see if we have any pieces lying around. (http://www.ultracorinc.com/ccarbon/ccarbon_body.html
3.) Most of the sandwich panel failures I have seen were due to buckling. The problem with sandwich panel buckling is that it is notoriously difficult to accurately predict. There are a multitude of failure mechanisms, which need to be analyzed. Face sheet wrinkling, disbonding, core crushing, crippling, ect ect. There are several NASA/NACA, DOT, and FAA reference papers out there, unfortunately they are only accurate for flat panels, with balanced laminates, under compression.(exaggerated) How actual sandwich structures fail is somewhat of a trial and error sort of progression.
That being said, if you have unlimited resources, an office full of PhD's, and a ton of money, I would look at creating a micro-mechanical model of the laminate in question. There are a few new programs out here which show some promise (GENOA is one) but the industry rumor is that in order to get this analysis to work, you have to do so much material/subcomponent testing, that you might as well have built and tested the real thing