In standby. The TV supports CEC so would turn on when asked.
Yeah, but in practice CEC has been iffy on most TVs i've owned, and i've never seen a TV that can startup in less than 10 seconds.
Plus AFAIK CEC can't turn back off the TV, so after the first use it's just on forever until you manually turn it off.
My Apple TV definitely turns my TV off via CEC, it works great
New AppleTV 4th gen has an IR blaster that'll turn your TV on and off even if your TV doesn't support CEC. It seems to figure out the right IR codes automatically.