Car keys usually remained powered off until a button is pressed.
Not saying this is what Tesla do. For privacy reasons it would be way better to not transit until it sees the car. But battery life isn't proof of this.
The expected distance is pretty high. Officially up to 10m (like a keyless entry) but they can reach much higher distances. Because I often leave my AirTag in the house and it gets detected very often despite me not even having an iPhone :)