Yes. Look at how many calories professional chess players are burning by just sitting and playing chess. You will be shocked.

Heck yeah it do. The brain is powered by glucose, and more brain activity will use much more of it. Jokingly, it's how I can tell a new engineer from one that is extremely experienced: A bright young engineer is usually skinny from problem solving all day. An older one is likely stuck in more meetings where brainpower is a liability and is probably on the heavier side.

Also, as an occasional eater of magic mushrooms, I keep packs of glucose in the house that are typically used for diabetic emergencies. Psilocybin pushes a brain into overdrive and causes my blood sugar to nosedive. (There are studies about possibly using psilocybin as a diabetic treatment to improve pancreatic function, btw..)