Cranking up the heater in cold should not reduce the mpg because heaters use the hot coolant (same coolant that circulates in the engine to draw the heat out of it) circulated in the blower. Cold starting engnines does use some extra fuel, until the engine gets back to normal operating temperature.
In summer AC does kill the gas mileage, because it needs a compressor to be driven, which takes power off of the engine. You can hear the change in the sound coming off from engine when you run the AC, you wont hear any of that while running the heater.
Are you sure it's a corolla and not a tundra? You said that the logo in front came off, so wondering if someone sold you something else calling it corolla :(.  Have a mechanic/dealer checked it to make sure it is what you think it is.