Does the brain and the body use the same energy source (glucose/ATP)?
You can grab any university level text on human physiology and you will find that the brain is a unique organ for cellular respiration. The brain is only capable of using two metabolic subtrates: glucose for oxidative phosphorylation (producing ATP) and ketone bodies.

Exercise does indeed divert blood flow toward the muscles. After all, they are undergoing higher levels of respiration and need nourishment. Your body will always balance muscular needs and the needs of the brain and the rest of the body. Similarly, digesting food has a similar response, although this is more involved. Eating food has a autonomic reflex to increase blood flow to the stomach and digestive tract to facilitate the processing and distribution of nutrients.