The brain only conserves 15–25% [1] of energy during sleep and does not appear to be a reason sleep evolved. It seems that the pressure between the cerebral spinal fluid and the interstitial fluid cannot build up to the required level while the brain is awake and fully active.
He's not talking about the brain conserving energy (the brain doesn't use that much energy anyway) but rather the organism as a whole by not performing other physical activity (e.g. an early human looking for food during the night when the African predators are active.)
Well, IIRC the brain receives 25% of the body's blood flow. I think that means it's using a lot of energy (but I'm not a doctor, or a physiologist, or a biologist...)
I think that's true. Here's more support for the idea that the brain actually uses quite a lot of energy:
"Although the average adult human brain weighs about 1.4 kilograms, only 2 percent of total body weight, it demands 20 percent of our resting metabolic rate (RMR)—the total amount of energy our bodies expend in one very lazy day of no activity."
The figure I've seen is 20%. If you burn about 2000 calories a day that works out at about 97 watts on average. 20% of that is about 19 watts. Bit more detail here:
Despite all of the progress in integrated circuits, the best we have is still nowhere near theoretical limits for computation per unit energy or unit space.
[1] http://jap.physiology.org/content/70/6/2597