Questions about blower!

We will need more info to be able to help. What stove is it ?

My blower had a magnetic sensor that attached to the underside of the stove. Long after the fire burned down, the coal base was always still hot and so was the stove metal, the fan would never really cut off unless the stove went completely cold - which may take more than 24 hrs. There was a heat shield underneath the stove as well and if I moved the magnetic thermostat to the heat shield, it would cut off - but it also took forever to come on. I know I had the sensor in the right spot on the stove because there was a round cut out specifically for the sensor and that's where the installer placed it. If you want it to cut off earlier, perhaps move the sensor to another spot - like the back corner? Or between the stove and heat shield? But if you do, it won't come on as easily.

I don't think the fan would pull basement cold air up. Mine had the intake below the stove and the output was above. So it would pull in floor level air in and heat it. The Heat output will rise pushing any cold air to lower floors, but it should only be drawing floor level air from the vacinity of the stove.

We ran our fan medium to high all the time. Sometimes lower if we were watching TV. But while sleeping - and for heat, we had ours on high.