Dashboard display and transponder

[AlanSilberlight]

My wife and I have a 2012 Yaris. We have noticed the following:
When we shut the motor off, two items remain on the display—the time and the total mileage. The gas gauge stays at the same level it was on before shutting the car off.

After leaving the car, if we press the lock button on the transponder, everything in the display turns off, the gas gauge goes to the empty mark, the alarm is set and the doors lock. If we do not press the lock button on the transponder, after a while the display turns off, the gas gauge goes to the empty mark, the alarm is set but the doors do NOT lock.

Is this what the car is supposed to do? Also, is there a possibility that the car’s battery will be drained if the display remains on?

Thank you very much.
Alan and Linda Silberlight