Putting your laundry issue aside, which is a problem all of it's own, its a fact.
OK, if we must, then we must.
CO2 on the other hand has contributed 2 watts per sq metre in total, not continuously, but in the history of the earth.
Source us the "fact" that in entire history of the planet a metre squared of CO2 "contributed" a total of 2 watts of energy. I'm guessing they mean per square metre of the earth's surface area - the amount of energy retained and released by CO2 - following the second law of thermodynamics after absorbing infrared radiation reflected from the earth's surface, which as you noted, was warmed by energy from the sun.
What's confusing me is the claim that the total amount of heat energy (2 watts) radiated as per the 2nd law by CO2
OVER THE ENTIRE HISTORY OF THE PLANET - not continuously - is 2 watts. First, how would you measure this over 4 billion years - probably less given we didn't have atmosphere for part of this 4 billion year period? Second, the amount of CO2 in the atmosphere has varied over time, as climate change deniers often point out - so again, how could this total amount of energy be measured or even estimated? I guess you could somehow do it with a model if you had enough data, but we know climate deniers don't like models.
Lastly, the energy total for CO2 seems a bit low - we know the sun provides around one kW/h per hour for every square metre of Earth during sunlight hours. Do you mean 2 W/h per hour maybe? The NASA figure I found in terms of heat energy imbalance (yes we know you dudes don't trust NASA) was 0.85 W/h per hour which doesn't sound much, but of course has a great cumulative effect over years.
Maybe just give us the source of this malarky so we can have a look for ourselves.