Based on the description on their site, the controller includes a built-in battery: "8.39 Wh Li-ion battery, 35+ hours of gameplay... "
That was disappointing for me. Specially condidering the Steam Frame's controllers make use of AA batteries: "One replaceable AA battery per controller, 40hr battery life"
AA Batteries might not be as convenient to use, but being able to replace them is a great advantage. All my Xbox360 controllers still work fine, but none of my PS3' Dualshock 3s.
The official docking station could be used to recharge (rechargables) AA batteries so the functionality could remain the same.
Definitely not. Even if no device is plugged in to a charger, the circuitry in the charger still uses electricity. Even a basic transformer with no control processor will still use electricity because they, like all machines, are not 100% efficient.
You're right, but the power is so low that you won't see anything using an ammeter