They don't transmit continuously. The little CR2032 cell in my Tesla key would be dead in a day or two but in fact it lasts about at least six months, probably more like a year. I presume that they listen for a ping from the car and only then transmit. Unless you press a button of course. I've had the car almost five years and I think I have only replaced the battery four times.
An AirTag transmits constantly yet it manages to last 6 months too. They do not wait for phones to ping them, they just beacon all the time.
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.
All the time, as in several times per second. I have a bluetooth sniffer and these beacons are very very chatty. I didn't test the AirTags myself but it can't be lower than once every 5 seconds or so. But really I think they are like the other battery-operated BLE beacons with several times per second. BLE really is really low energy, they got that part totally licked.
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 :)
