If I have my thermostat set to any temp for a long time, then turn it off because it's getting too hot or too cold in the car, air is still blowing through the floor vents. I could have it blowing through the defrost or dash vents only, but once I turn it off, air continues to blow through the floor vents while I'm driving. And it blows at whatever temp I had it set too before turning it off. This continues until I come to a stoplight; when the light turns green, it starts blowing once I move again. Is this normal or what?

 

Yes it is normal.
While ac is off, push hte mode button - you will notice that hte air that was once comming through the floor vents now comes through your selection. If you want it to stop, push hte air recirculating button.