I am restricting my consideration, for the moment, to just the troposphere. With regard to our understanding of the GHE do we predict that an increased GHE due to increasing CO2/Water/Methane will change the lapse rate postively or negatively? The dry adiabatic lapse rate (DALR), according to my old text books, is easily calculated from the formulae -g/c, where g is the acceleration due to gravity and c is the specific heat capacity of the atmosphere. This is supposed to be a general formulae that can be applied to any planet with a solid surface and a gaseous atmosphere. So for earth is the formulae now -g/c + (GHE) or is it -g/c - (GHE)? The bracketed GHE term being specific to the atmosphere of a specific composition, e.g. including 400ppm CO2. The second part of the question then , after we have the correct sign, is how is the term (GHE) expected to vary as the CO2 (or other greenhouse gas) concentrations vary?
For dry adiabatic, you simply ignore the greenhouse gas. This might sound like a cheat, but it's not. The dry adiabatic lapse rate doesn't change with temperature or latitude.
For a physical explanation, if you consider what the greenhouse gas does, it captures heat and raises the temperature near the Earth's surface, which expands the troposphere and raises the tropopause. In other words, the troposphere grows taller, so the lapse rate can remain the same and the tropopause moves a little higher up. This can also be seen in higher and lower latitudes on Earth in that the tropopause is about 5 miles closer to the Earth at the poles than over the equator.
The adiabatic lapse rate is a property of the ideal gas law. Heat trapping greenhouse gas doesn't affect it because the gas expands accordingly with trapped heat.
The true environmental lapse rate is a bit more complicated as that fluctuates with humidity. As the Earth warms, absolute humidity should increase but relative humidity should slightly decrease. A lower relative humidity, on average, should raise the lapse rate slightly (I think), as the dry adiabatic rate is greater than the moist rate. But this last paragraph take with a grain of salt. The environmental lapse rate is more complicated.
2 identical planets, with 2 identically thick atmospheres, same c, same g, same lapse rate but one is a pure CO2 atmosphere and the other is a non greenhouse gas with the same c (fictional gas, we will call it CO2U)
I'm not an expert in lapse rate, so my answer is a little bit of a cheat, but I'll take the thought experiment as best I can.
Fictional gas - same density as CO2 (at standard temperature & Pressure "STP"). Both atmospheres are on planets of identical surface area, with identical gravity and both atmospheres have identical mass and (just to clarify) Identical density at STP.
The atmosphere that traps heat will expand, so it will have a higher troposphere. Hot gas expands. Cold gas contracts.
So the planet without the greenhouse gas will have a troposphere that's 6 miles high, the one with the greenhouse gas and a higher surface temperature will have a troposphere 10 miles high. The rate of temperature drop can remain consistent at (-g/c) but the hotter planet will have a higher troposphere and a more expanded atmosphere. That's for the theoretical model anyway. I don't do real world models. Those are much harder.
So the planet with the GHE atmosphere will be hotter at the surface, hotter at any given height in the atmosphere and hotter as a complete body/system (atmosphere+solid planet). Everything is hotter. Errr... what happened to the supposed radiative S-B balance as viewed from space then ?? Radiative input from sun the same, radiative output from the GHE planet now higher
The energy balance question is a fun one, especially when climate change "skeptics" use it to say "climate change can't possibly be right" - look at the chart they use, more heat leaves the Earth's surface than comes from the sun.
Lets ignore that debate for brevity and look at the overall theory of increasing greenhouse gas driven climate change. The Energy from the sun is same, so, why does Earth get warmer? The answer is trapped heat. So, what does this look like from space? Counter intuitively, from space the Earth looks colder than it did 30/40/50 years ago. It's giving less thermal radiation into space. This is because heat is being trapped in the lower atmosphere and oceans. The upper atmosphere on Earth is actually cooling and contracting and less thermal radiation is leaving the Earth, not more, even with the warming surface temperature. Measuring the decrease in heat that leaves the Earth is one way to measure the total amount of heat being trapped by climate change.
In time, a new equilibrium will eventually be reached and heat in will equal heat out again but for now on Earth, Heat in is greater than heat out. The difference is heat added, which does things like warm the air, the oceans and melts the ice. In your scenario, there's no change, it's already warmer, but it's warmer in equilibrium, so heat in = heat out.
In other words: A warmer atmosphere at equal heights doesn't mean more heat leaves the planet, because the CO2 rich atmosphere makes it harder for the thermal radiation from the warm surface to reach space. The greater heat is balanced out by a greater percentage being reflected back to the surface. The amount that goes into space is equal on the 2 planets.