(Hi Jack ...) Well, kc, you're in a tough spot. I have to ask how hot does it get right now in the attic? And if you don't have an eaves or soffit, do you have gutters? I too have never seen a home w/o an eaves. Well, I have but their roofs were FLAT.
The 'common' reason attics are to be vented is for temperature equalization (outside and inside). Also, yes, there are moisture issues. For cathedral ceilings, the most common problem is mold and mildew in the cathedral part of the ceiling. For attics, it's trapped heat and ice damming. There are just too many problems if the attic isn't vented somehow. The same for cathedral ceilings.
You're right, Jack, about the walls not being vented. But heat rises and eventually ends up in the attic. Once it's there, where DOES it go? It conducts through the under-sheathing and then out (assuming it's cold outside.) In the summer, the opposite happens because the roof (on the outside) gets much hotter than the air in the attic. At some point, the temps will equalize and then the heat ends up getting 'pushed' into the living space. NOW, you have a problem!