2) I read elsewhere (I can only research what I read, I don't really have the ability to check much of this for myself) that models assume a constant lapse rate. Chris said the lapse rate is required for the greenhouse effect, but from everything I look at people only catgorize in "Dry" or "Moist" cases, but doesn't it vary everywhere over the globe?
There are models, somewhere, that assume anything one could mention, so I suppose there are some which assume the lapse rate. As you correctly notice, though, lapse rates depend on conditions, and those conditions vary over the globe. A serious climate model couldn't assume the lapse rate. And, in truth, they don't. More in a moment, but something to look back at is my description of the 16 climate models
Let's start with the lapse rate itself. What it is, is the change in temperature with elevation. Through the troposphere, the lapse rate is a negative number (cooling with elevation). In the stratosphere, it turns to zero and then positive (warming with elevation). In the mesosphere, we go back to cooling with elevation. This is a strictly observational issue. You can find temperature profiles, say from the Standard Atmosphere (a specific thing; Project: take a web look for it and see what they look like; they've changed through time, by the way). And then find the temperature difference between two levels, and divide by the elevation difference. That'll give you the average lapse rate. You can also find, radiosonde soundings of temperature. (I'd start my search for this project at the National Climatic Data Center.) This will let you see how the lapse rates vary day to day at a location, and between locations.
On the theoretical side, we go back to Conservation of Energy. We start with a completely dry (meaning no water vapor) blob of air, in an insulating bag that prevents it from radiating, conducting, or convecting energy to or from the surroundings. Then we lift it through the atmosphere. As we do so we'll find that its temperature drops. This happens because our blob does work in expanding. The energy for that work comes from its own thermal energy store. We can compute exactly how much the air would cool under this circumstance. It is about 10 K per km near the surface of the earth. This is what we are referring to in talking about the Dry Adiabatic Lapse Rate. The 'adiabatic' refers to our insulating bag around the air blob.
The polar regions, particularly the Antarctic plateau, are not bad approximations to that situation. But most of the atmosphere has fairly significant amounts of water vapor. We start, now, with a slightly different bag. It still prevents heat to be added or lost to the bag from outside. But now there's a second energy source inside the bag. Water vapor can condense, and when it does, it will release energy. We take the approximation that all the heat energy goes to the gases in the bag, and that the newly-formed liquid water is immediately moved outside the bag.
Now when we lift the bag, things go a bit differently. Let's start with air at 70% relative humidity, a typical global mean value. As we lift the air, it first acts 'dry', so cools at the about 10 K per km rate. But after a while, we have cooled to the point of being at 100% relative humidity. When we start to lift any further, water starts condensing and releasing heat. The condensation only happens if we're still cooling, so it can't reverse that tendency. But it can greatly slow the rate of cooling. This gives us a Moist Lapse Rate. Note that I dropped 'adiabatic' from the description. Since material is leaving the bag, it isn't an adiabatic process any more. It is pseudoadiabatic (a term you'll see) -- almost adiabatic, as the loss of mass isn't large. But not entirely adiabatic.
As a typical ballpark value, we take 6.5 K per km as the moist lapse rate. But this obviously will depend a lot on how much water was in the bag to begin with, and the temperature. If we start with a very warm, saturated, bag of air, then the lapse rate can be even lower than the 6.5 K per km. If we start, though, with a cold blob of air, even if it is saturated, we are still close to 10 K per km lapse rate. The thing is as we get colder, there's less water vapor present, which gives less condensation, then less heating. Consequently even in the tropics, the lapse rate heads towards the dry adiabatic value as you get high above the surface.
Whether moist or dry, the lapse rate computed this way is an idealization. In the real atmosphere, radiation does move energy around, and blobs of air do mix with each other (even when rising). Still, it's derived from a strong scientific principle (conservation of energy), and it turns out to give us good ideas (in reasonable accord with observation) about what the atmosphere should look like in the vertical.
For the modelling, let's think back to the 16 models. First, many of them are never used, so we'll ignore the longitude-primarily models. That leaves us with the 0 dimensional model I've already given an example of, and there's not even the opportunity to impose or even make use of a lapse rate in that. The 4 dimensional model definitely doesn't assume a lapse rate -- doing so would force violations of conservation of energy. Radiative-convective models can't force the lapse rate for the same reason. For a discussion of such models, to which I'll be returning in another post about water vapor's greenhouse contribution, see Ramanathan and Coakley, 1978. As of that era, one did specify a critical lapse rate. This isn't the lapse rate that the model had to have, rather, it was a limit. If the limit were violated, something has to happen. That something is to conserve energy by mixing the layers that violated the limit. And Energy Balance Models, as I expected, don't even mention lapse rate. See North, 1975 for a discussion of energy balance models.
Either the models are too simple to know about lapse rates (0 dimensional, Energy Balance), or they compute the lapse rate (Radiative Convective, 4 dimensional). Either way, the lapse rate is not assumed before hand. It's an interesting after the fact diagnostic for the Radiative Convective or 4d models, or impossible to speak to.
One thing to do is find some better sources for you to read. I taught an introductory (freshman level) physical geology class with Lutgens and Tarbuck, and liked the text there. They have a text at that level for meteorology, but I haven't read it myself. It should be good, though. John M. Wallace and Peter V. Hobbs, Atmospheric Science: An Introductory Survey is an excellent book. In half the chapters, comfort with multivariate calculus is assumed. But the other half are descriptive/physical rather than quantitative/mathematical so should be approachable already. A second edition is now out, I used the first. Does anyone have suggestions for a good freshman level introduction to meteorology/climate?