Then how would you explain thermosphere with only the facts you mention? It starts just below the Kármán line at 90km above the sea level, where the pressure is much lower than even the peak of Mt. Everest, and yet the temperature range within is between 500 and 2500 °C. You aren't saying anything that's wrong on its own, but it's not the full picture of the problem. Think of what happens in the atmosphere at different heights, its composition and how it affects the flow of energy.
Hmm, good question. That's really high up, far higher than aircraft travel. I would guess that there's an interaction with solar radiation of some kind. Is it where the magnetic field deflects the "solar wind" perhaps? With the atmosphere that thin and pressure so low, I do think most heat energy absorbed near the Earth's surface would be dissipated, continuing the pattern seen on mountaintops. Thinking about this abstractly, I realized I didn't know why the Earth's core is hot. If it was formed by a lot of space dust accreting, it would seem more likely to be very cold. And if the sun is the source of the heat, it should heat the surface more than the core. So I have to guess that the same mechanism is at work: the intense pressure due to gravity is enough to heat the core beyond iron's melting point.
Let's focus on one problem at a time. Also, let's not forget that every step was simplified. You can simplify the initial problem to the following: you have a source of energy (Sun), a black body screen (Earth's surface) and a transfer medium between the two that can act as a buffer. Since both of them radiate, there should be a point where energy going in, is equal to the energy going out. If you'd have two bodies at T1 and T2 (T1 < T2) and connect them with a strip of copper, you could reasonably expect there's going to be a point on the strip where T = (T1 + T2) / 2. That's your basic concept of heat transfer. Now, time for some facts regarding our buffer: The density and pressure of the atmosphere change with the distance from the Earth's surface. The amount of energy per volume that air can retain is proportional to the number of particles enclosed in said volume (and their energies). The average velocity of those particles is proportional to temperature and kinetic energy. Mass is another important factor, pertinent to the original question. Heavier gasses stay close to the surface while lighter ones go up. It affects the distribution of gasses throughout the atmosphere. All gasses in the atmosphere (at least up to at least 80-90 km) are composed of more than one atom. N2, O2, CO, CO2, H2O, O3, Ar, CH4, NOx. Those are the most important ones. I'm going to use the word 'particle' when referring to them. The amount of energy a particle can retain, and how hard or easy it is to change its temperature, is related to its heat capacity and specific heat. This has three components, each related to the type of motion a particle can make. First is your basic translational motion, which is to say "how a particle travels in 3D space". Up, down, left, right, forwards, backwards and their combinations. Second is the way a particle can rotate. Because we have only three axes (and three moments of inertia associated with them), it can only spin around X, Y or Z axis (or a combination of them). The third way a particle can move is the way it can vibrate. Imagine balls connected by a spring, and you have the basic idea. For an n-atom particle, there are (n - 1) ways it can vibrate. The more ways a particle can move, the number of degrees of freedom, gives it a broader range of energy it can absorb. Each of those plays their part due to the equipartition of energy. Broadly speaking, each degree of freedom gets proportional for its type share of energy. Meaning: they have more ways to absorb and emit radiation. FYI: Wikipedia has a good article on rotational-vibrational spectroscopy. I told you to consider what happens at various heights of the atmosphere. I did it to point you toward things like clouds and water vapour in general. They scatter light and absorb tremendous amounts of energy on their own. Combine it with all the greenhouse gasses, some mentioned above, that are most excited in the infrared spectrum, and you get even more buffer properties. They act like a one-way mirror, absorbing energy radiated from the ground and radiate almost all of it it back towards the ground. That's part of the reason (if not the sole reason, I'm not certain about the specifics) why satelite images of clouds are done in IR spectrum. Now, let's bring those facts together. Close to Earth's surface, you have more of those massive, multi-atom particles. This skews the point of energy transfer equilibrium closer to the surface. Wait, so where's the role of pressure? It's talked about indirectly. Pressure and temperature are related to a collective force given by all the particles hitting surfaces (translational motion). The higher you go and the fewer of those particles are present, the less energy they carry on average. There's a second part to this post that I'm still writing. It's more about convective processes and why/how they don't matter nearly as much as you might be thinking. My goal is to write it from the first principles approach like this one, but it's going to take me a while.
nah i think the other problem is more interesting than gases. one at a time is so '90s. witness: -it seems intuitive that random space dust which became the core is cold because space is cold... but remember space is cold because it's empty-ish and the core is very much not empty, so lots of heat got trapped in the core. there was, at the time, lots of jostling. jostling is heat. -as mr. devac pointed out, we can figure out how old rocks'n'stuff is sometimes by measuring radiation, which is just more heat -also the layers of the earth trap heat pretty well right, i mean there's a ton of magma down there and not many access points on the surface and the sheer magnitude of energy is unnerving-- temp of sun at surface 6000 kelvin. temp of earth, at same, a few hundred. massive difference in energy, so a sort of "arbitrage" occurs (obviously not really, and not forever, pace second law) and order springs from disorder. because order is just spending energy to create a temporary equilibrium. anyway, i don't know a damn thing about the thermosphere but the point is we don't really blast that much heat out. we trap it, and have since day 1, and we also accidentally used it to create a bunch of neat shit. https://en.wikipedia.org/wiki/Autocatalysis#Creation_of_order borrowed time, but then we're in good company -- the whole universe is borrowing tooLet's focus on one problem at a time.