skinnier tires certainly have a disadvantage, less traction! Traction is also related to the tire compund and tread pattern, but the less of it you have on the road, the less traction you are going to have. So I wouldn't go too skinny, its better to not loose control of your car.
I definitely don't know everything or close to it about this subject, so keep on your quest!
As for your diameter question, I don't know. You could get the weight measures for a certain tire and wheel setup and see which weighs the least. Let us know, I am curious. I have done a little of that myself, but I forget what the result was.
Not sure about your hunch other, maybe someone else is.
Good luck!
