Do different activities require more energy supplied or is it more or less a constant demand?
I’m sorry, I cannot answer this question entirely, but energy requirements for the brain do vary considerably on a small-scale as activity paterns change. This is the basis for fMRI imaging; basically, as a certain part of the brain becomes more active blood-flow to that area increases (actually overcompensates) and this is the basis for the contrast of a functional MRI scan.
This is on a sort of 10-15 second timescale; I cannot provide a good answer for longer time scales or tell you if it’s calorically significant or if so by how much.
My understanding is that during a coma, the brain uses far less oxygen and blood sugar. I don’t have a cite, though.
Many years ago I did a stint in a neurobiology research lab. This was when brain imaging with FMRI to detect brain activation was a pretty new thing. It has probably progressed quite a bit since then.
The researcher I designed equipment for told me that they were looking for changes on the order of one half of one percent in blood flow for the parts of the brain that were being activated.