The lapse rate is actually pretty complicated so let's go through it ...
The basic explanation for the lapse rate involves modelling a parcel of air rising in our atmosphere. When you do this involving dry air, and you assume the atmosphere is in hydrostatic equilibrium (the flow velocity is constant over...
Best answer: The lapse rate is actually pretty complicated so let's go through it ...
The basic explanation for the lapse rate involves modelling a parcel of air rising in our atmosphere. When you do this involving dry air, and you assume the atmosphere is in hydrostatic equilibrium (the flow velocity is constant over time), you can derive an equation which tells you the lapse rate is dependent on only the specific heat capacity of air (about 1000 J / kg K) and the acceleration due to gravity (about 9.8 m/s^2). Dividing 9.8 m/s^2 by 1000 J/ kg K and you get 9.8 C per km, or a drop of about 1 degree for every 100 metres you climb.
This is complicated by the addition of water vapour. If you consider a parcel of moist air, you now have to account for the latent heat of vaporization of the water and how the drop in temperature affects that water vapour. To put the equation in simple terms, let's imagine you have a parcel of moist air rising. At some point you'd have saturation, and as the parcel gets cooler still you'd have condensation of that water vapour, producing a cloud which extracts heat from the parcel of air. Since this is temperature dependent there's no fixed lapse rate but a reasonable value could be a drop of 5 C for each km of height gained. However, it's important to note that what you'd actually see is a transition - the air would follow the dry lapse rate before saturation, then start to follow this moist lapse rate afterwards.
Now, you can keep adding complexity to try to get better models but the point is that these describe the lapse rate in the troposphere where temperature drops as you increase in altitude. As you move into the stratosphere the exact opposite happens, and temperatures increase as you increase in altitude due to the effect of ozone. As I said, the lapse rate isn't simple and it isn't just due to gravity.
In any case, let's imagine you have a warm troposphere which decreases in temperature as you increase in height and a colder stratosphere that increases in temperature as you increase in height. The temperature of the troposphere is ultimately dependent on how much energy it radiates out into space. As we increase our CO2 emissions, the temperature of the troposphere at the equator would increase but that increase in temperature occurs higher up in the troposphere, resulting in more energy being radiated into space. This reduces the lapse rate since the troposphere is warmer higher up (the lapse rate is large if there is a large temperature gradient between ground and top of troposphere) and represents a negative feedback which reduces the impact of global warming. As you move towards the poles, that warming occurs lower in the troposphere and increases the lapse rate, which represents a positive feedback. What happens at the tropics tends to dominate, so what we see is a reduction in the lapse rate and a negative feedback on warming.
These processes are included in the IPCC models.
Additional: Just in response to your update ... yep. That's one way of looking at it. The other way is, if you imagine you have a warmer troposphere, that means as you increase the altitude of your parcel of air, it doesn't expand by as much as with the cooler troposphere because the warmer atmosphere exerts a higher pressure on it. This balances out so you get the same lapse rate. Remember, the dry adiabatic lapse rate just tells you if you have a certain gas and a certain gravitational pull, what the temperature decrease would be as a function of height. It doesn't care what the initial temperature is. So cold or hot atmosphere, you'd get the same lapse rate.
BUT, as I said in my original answer, this isn't how the real atmosphere behaves. It's a reasonable approximation. The lapse rate is the end result of processes occurring in the atmosphere (of which the change in pressure as a function of altitude is one factor). Global warming must change the lapse rate if the troposphere warming is not evenly distributed throughout it. This factor isn't included in the simple model of lapse rate because that process is modeled as being adiabatic - there is no heat loss from the modeled 'parcel' of air to its surroundings.