A faulty thermostat could indeed lower your fuel mileage. Change it (with a good name-brand replacement) and hopefully your heat will be good again. Also check for a plugged heater core.
Winter gasoline and summer gasoline are really two different fuels. Winter gas is blended for easier starting and has fewer BTUs than summer gas. If your mileage is down by a couple MPGs (or even more), winter gas could be one reason why.