As long as you haven't bypassed the throttle body heater line (the coolant line that goes through the throttle body, what ever it is called) your car will be fine in the cold. It keeps your throttle from freezing. The air entering the engine won't be -20 by the time it gets into the cylinders.
Rodding around before it is fully warmed up will be alot more hurtful than just extreamly cold air.
Not quite. The engine is already in the cold. It is the internal heat that is building up for the temperature change. Everything starts at the same temperature (relativly in some cases, depending how fast the weather changed or how long the vehicle has been sitting blah blah blah) so you are not going to be "splashing cold water" on the engine, since the engine has been imersed in the temp since before start up.