"OK, I can live with that, - if you can accept that the sun’s energy input to Earth and the radiation output from Earth has been in balance for millions of years."

If you regard swings in global temperature of more than 10 C "in balance" and rises & declines in sea levels of around 100 meters "in balance". Shall we go into the part where most of the continents are covered with mile thick ice caps as also part of this balance?

"Heat is kinetic energy, or motion of molecules."

Ok, it was the "accumulation" part which was causing me to wonder, but I guess it's the same as getting warmer. But I was thinking about it after I posted, and guess it's *possible* for some parts of the atmosphere- particularly above the Stratosphere for the kinetic energy to remain for longer periods of time [so I withdraw my assertion about it being for periods of scale of seconds or nanoseconds] I simply don't know the duration this is possible- though months to years seem unlikely.
In other words if you had some sphere which was made of glass which was transparent to infrared radiation which is leaving the sphere but not transparent to any incoming infrared or heat from conduction from outside [or heat conducted from inside]. And the surrounding air was cold and low pressure, and the air inside the sphere was also low pressure and cold but say 10 C warmer than outside air- how long would it take to lower the temperature by say 5 C?
Of course, the gas molecules would collide with walls of sphere, so the larger the sphere the more it would simulate what happening in the atmosphere- in regard to accumulation or storing of kinetic energy.

"In other words; the alpha, beta, and gamma radiation from nuclear reactions can increase the temperature of surrounding molecules, by imparting kinetic energy to the molecules.
These energetic molecules then cause the material to become hot."

Well, in that case if you talking about the troposphere- the heat will be transfered rapidly via convection and conduction and of course radiate energy at speed of light. And globally there is no evidence that there is a higher levels of "alpha, beta, and gamma radiation" than compared to other times in last million or billion years.
And as I said before, all you would need to test this idea is take the temperature of the air say within a mile from a nuclear reactor and compare this to similar location which doesn't have reactor within 1000 miles.

"The higher the kinetic energy [or temperature] of the molecules are, the hotter the material becomes."

And obvious the hotter the material the more energy it radiate, conducts and is transported away by warming air.

"The planet and its atmosphere is a type of a heat engine, see:

http://www.columbia.edu/itc/ldeo/v1011x-1/jcm/Topic3/Topic3.html

If we add more kinetic energy to the atmospheric molecules, the atmosphere can do more work in the form of more destructive storms."

That article is about geology and in the interior of earth- rock or brick is a good insulator- a mile of rock works well at insulating any heat.

As for destructive storms- it has more to do with temperature difference than how hot something is- Venus isn't particular excessive in regard to violent storms- though it's cooler poles make quite a hurricane:
http://news.softpedia.com/news/Venus-Mysterious-Haze-Replaced-by-Violent-Hurricane-80851.shtml

And on Venus generally speaking you have high winds at higher elevations racing towards the unlit side of the planet.

But Neptune has highest wind speeds of any planet in the solar system with "Mean cloud temperature -193 to -153°C "
http://www.solarviews.com/eng/neptune.htm

So have high gravity and big differences in temperature and you will get violent storm. Though of course the Sun isn't a calm place.