Are you mistaking different for correct? Generally the thermostat just turns the pump on, but is there actually anything wrong with leaving the pump on and changing the temperature?
Yes: efficiency and keepimg temperature stable at a certain point (ie, with a small hysteresis).
Ad 1: the gist is that it takes less fuel to heat a liter of water from 30 to 31 degrees, than it does to heat it from 60 to 61 degrees.
Ad 2: with a lower temperature, it becomes easier to control your overshoot. Actually this is related to 1, but what you want in the end is to have a system that runs at as low a temperature as possible, all of the time.
I'm not entirely sure, however running the pump without actually heating the water seems like a waste of energy. Maybe that's how this boiler works though, I'd be interested to know more from the OP.
