Opening the fridge door to cool a room

No, you are making the fridge do extra work, so more energy is coming in (through the plug) as the pump continues to run since it's not reaching it's cold point. A normal operating fridge does not manifest cold air; it just pumps all the heat out of the inside of the fridge.

The action of pumping the heat out also has heat as a byproduct (which is waste heat from the power coming through the plug as it does work with the pump). So you're really just generating more heat and moving heat around.