That's pretty normal behavior for a TV because imagine how annoying it would be if it shut off during a layer change of a DVD or otherwise whenever the screen goes briefly blank. Some brands do shut off automatically if no signal is detected for some time.
As for the green LED, I'd imagine it's like the single green LED on many motherboards attached to +5vsb that's there to remind you that it's powered, so that you should unplug it from the wall before attempting to remove any cards (in the case of PCIe it's indicating the presence of +3.3v standby). Usually it's a red LED that indicates a fault such as the power connector hasn't been plugged in