"How much coal has to be burned to heat the earth 1 degree?"
Interesting question!
Naively, 1 kg coal ~ 3*10^7 J when burnt (link: https://www.euronuclear.org/info/encyclopedia/coalequivalent.htm). The specific heat of the atmosphere is about 1158 J/kg*C and there's about 5.14*10^18 kg of atmosphere, so the total heat capacity of the atmosphere is about 5.95*10^21 J/C. Using similar calculations, the oceans' heat capacity is about 5.30*10^24 J/C, which dominates the calculation. If we burned about 1.6*10^14 kg of coal and somehow achieved perfect heat transfer to the atmosphere and oceans, we'd have increased the temperature of the atmosphere by 1 degree, but the ocean would absorb it. It would take about 1000 times that amount of coal to increase the temperature of the atmosphere AND the ocean by 1 degree. We burn about 8bn tons of coal per year (link: http://science.time.com/2013/01/29/the-scariest-environmental-fact-in-the-world/), which is about 7*10^11 kg. So I guess we aren't burning enough to heat the atmosphere. Phew! (Well, there's all the gasoline and oil and natural gas and wood and so on...but I'm sure they don't matter.)
Of course, then we get into force multipliers. The primary effect of burning coal on Earth's temperature isn't actually the heat generated by the reaction, but rather the effect of the waste products on Earth's absorption of solar radiation. As you have astutely (and frequently) noted, the Sun is where most of our thermal energy comes from. But it's one thing to be bombarded with photons and quite another to capture the resulting heat. (This is the primary reason why the Moon is much colder than the Earth despite receiving the same amount of solar energy per unit surface area *and* having a lower albedo then the Earth.) Atmosphere is what captures the heat the Earth radiates after absorbing all those photons. So you see why the gases in the atmosphere might be important.
If you burn enough coal to pump ~4bn tons of CO2 into the atmosphere per year over 40 years (170bn since the '70s per link: http://cdiac.ornl.gov/trends/emis/tre_glob.html), plus whatever other side products might influence matters, it's enough to change the amount of CO2 in the atmosphere by about 20% (link: http://www.esrl.noaa.gov/gmd/ccgg/trends/). That isn't a large amount relative to total atmosphere, but the interesting thing is that CO2 absorbs heat from a different spectrum of radiation than water vapor--importantly, the 12-15 micrometer band, which is near the center of Earth's black body radiation spectrum. So this change is enough to have an effect, albeit not a large one.
But that triggers another force multiplier, because it upsets the water vapor equilibrium. People talk about water vapor being the dominant greenhouse gas, and they're right, but pumping water vapor into the air doesn't do anything because it just condenses right back out as rain. But raising the temperature by a small amount means a higher maximum amount of water vapor in the atmosphere, which raises the temperature again, which raises the maximum again...this feedback loop decays reasonably quickly, but the result is an extra rise in temperature.
This is about the limit of my knowledge on the subject, and there are a billion complicating factors I haven't addressed, so I won't bother taking a strong AGW stance one way or the other. But I hope this discussion at least addresses some of the basic issues like why burning coal can influence, not the amount of solar energy we *get*, but the amount of solar energy we *keep*.
A special note for a special inanity: None of this requires that natural climate change does not happen. As BOG frequently notes, man did not cause the last ice age; but the fact that nature caused an ice age in the past does not mean CO2 and other byproducts of human existence aren't influencing climate in the present. If you never bring up that argument again, it'll be too soon, BOG.