Stirring the coffee involves a transfer of energy from you to the coffee. That's because you are doing physical work on the coffee by pushing it around as it moves in the direction of your push. What began as chemical energy in your body becomes thermal energy in the coffee. That said, the amount of thermal energy you can transfer to the coffee with any reasonable amount of stirring is pretty small and you'd lose patience with the process long before you achieved any noticeable rise in coffee temperature. I think that the effect you notice is more one of mixing than of heating. Until you mix the milk into the coffee, you may have hot and cold spots in your cup and you may notice the cold spots most strongly.
When you put fans in front of the vents, you are probably causing the air conditioner to pump roughly the same amount of heat out of the room air as it would at 75 °F without the fans. As a result, the fans probably aren't making the air conditioner work less and aren't saving much electricity. In fact, the fans themselves consume electricity and produce heat that the air conditioner must then remove, so in principle the fans are a waste of energy.
However, if the fans are directing the cold air in a way that makes you more comfortable without having to cool all the room air or if the fans are creating fast moving air that cools you via evaporation more effectively, then you may be experiencing a real savings of electricity.
To figure out which is the case, you'd have to log the time the air conditioner cycles on during a certain period while the fans were off and the thermostat set to 75 °F and then repeat that measurement during a similar period with the fans on and the thermostat set to 78 °F. If the fans significantly reduce the units runtime while leaving you just as comfortable, then you're saving power.
Heater-based refrigerators make use of an absorption cycle in which a refrigerant is driven out of solution as a gas in a boiler, condenses into a liquid in a condenser, evaporates back into a gas in an evaporator, and finally goes back into solution in an absorption unit. The cooling effect comes during the evaporation in the evaporator because converting a liquid to a gas requires energy and thus extracts heat from everything around the evaporating liquid.
The most effective modern absorption cycle refrigerators use a solution of lithium bromide (LiBr) in water. What enters the boiler is a relatively dilute solution of LiBr (57.5%) and what leaves is dense, pure water vapor and a relatively concentrated solution of LiBr (64%). The pure water vapor enters a condenser, where it gives up heat to its surroundings and turns into liquid water. To convert this liquid water back into gas, all that has to happen is for its pressure to drop. That pressure drop occurs when the water enters a low-pressure evaporator through a narrow orifice. As the water evaporates, it draws heat from its surroundings and refrigerates them.
Finally, something must collect this low pressure water vapor and carry it back to the boiler. That "something" is the concentrated LiBr solution. When the low-pressure water vapor encounters the concentrated LiBr solution in the absorption unit, it quickly goes back into solution. The solution becomes less concentrated as it draws water vapor out of the gas above it. This diluted solution then returns to the boiler to begin the process all over again.
Overall, the pure water follows one path and the LiBr solution follows another. The pure water first appears as a high-pressure gas in the boiler (out of the boiling LiBr solution), converts to a liquid in the condenser, evaporates back into a low-pressure gas in the evaporator, and finally disappears in the absorption unit (into the cool LiBr solution). Meanwhile, the LiBr solution shuttles back and forth between the boiler (where it gives up water vapor) and the absorption unit (where it picks up water vapor). The remarkable thing about this whole cycle is that its only moving parts are in the pump that moves LiBr solution from the absorption unit to the boiler. Its only significant power source is the heater that operates the boiler. That heater can use propane, kerosene, electricity, waste heat from a conventional power plant, and so on.
The atmosphere maintains a natural temperature gradient of about 10° C (which is equivalent to 18° F) per kilometer in dry air and about 6 or 7° C (which is equivalent to about 12° F) per kilometer in moist air. The higher you look in the lower atmosphere, the colder the air is. Because of this gradient, it may be 20° C (68° F) in the valley and 0° C (32° F) at the top of a 2,000 meter high mountain.
This temperature gradient has its origin in the physics of gases—when a gas expands and does work on its surroundings, its temperature decreases. To see why this effect is important, imagine that you have a plastic bag that's partially filled with valley air. If you carry this bag up the side of the mountain, you will find that the bag's volume will gradually increase. That's because there will be less and less air overhead as you climb and the pressure that this air exerts on the bag will diminish. With less pressure keeping it small, the air in the bag will expand and the bag will fill up more and more. But for the bag's size to increase, it must push the air around it out of the way. Pushing this air away takes work and energy, and this energy comes from the valley air inside the bag. Since the valley air has only one form of energy it can give up—thermal energy—its temperature decreases as it expands. By the time you reach the top of the mountain, your bag of valley air will have cooled dramatically. If it started at 20° C, its temperature may have dropped to 0° C, cold enough for snow.
If you now turn around and walk back down the mountain, the increasing air pressure will gradually squeeze your bag of valley air back down to its original size. In doing do, the surrounding air will do work on your valley air, giving it energy, and will increase that air's thermal energy—the valley air will warm up! When you reach the valley, the air in your bag will have returned to its original temperature.
Air often rises and falls in the atmosphere and, as it does, it experiences these same changes in temperature. Air cools as it blows up into the mountains (often causing rain to form) and warms as it flows down out of the mountains (producing dry mountain winds). These effects maintain a temperature gradient in the atmosphere that allows snow to remain on mountaintops even when it's relatively warm in the valleys.
Unfortunately, the answer is no. The atmosphere is too complicated to be described by a simple formula or equation, although you can always fit a formulaic curve to measured pressure values if you make that formula flexible enough. The complications arise largely because of thermodynamic issues: air expands as it moves upward in the atmosphere and this expansion causes the air to cool. As a result of this cooling, the air in the atmosphere doesn't have a uniform temperature and, without a uniform temperature, the air's pressure is difficult to predict. Radiative heating of the greenhouse gases and phase changes in the air moisture content further complicate the atmosphere's temperature profile and consequently its pressure profile. If you want to know the air pressure at specific altitude, you do best to look it up in a table.
One of the principal observations of thermodynamics (and statistical mechanics, a related field) is that vast, complicated systems naturally evolve from relatively unlikely arrangements to relatively likely arrangements. This trend is driven by the laws of probability and the fact that improbable things don't happen often. Here's an example: consider your sock drawer, which contains 100 each of red and blue socks (it's a large drawer and you really like socks). Suppose you arrange the drawer so that all the red socks are on one side and all the blue socks are on the other. This arrangement is highly improbable—it didn't happen by chance; you caused it to be ordered. If you now turn out the light and randomly exchange socks within the drawer, you're awfully likely to destroy this orderly situation. When you turn the light back on, you will almost certainly have a mixture of red and blue socks on each side of the drawer. You could turn the light back out and try to use chance to return the socks to their original state, but your chances of succeeding are very small. Even though the system you are playing with has only 200 objects in it, the laws of probability are already making it nearly impossible to order it by chance alone. By the time you deal with bulk matter, which contains vast numbers of individual atoms or electrons or bits of energy, chance and the laws of probability dominate everything. Even when you try to impose order on a system, the laws of probability limit your success: there are no perfect crystals, perfectly clean rooms, flawless structures. These objects aren't forbidden by the laws of motion, they are simply too unlikely to ever occur.
Living organisms create more disorder in their surroundings than they create order in themselves. Overall the disorder of the combined system—organisms and environment—increases. This result is an unavoidable consequence of the second law of thermodynamics, which notes that the entropy (disorder) of an isolated system can never decrease. While it is possible in principle for a living organism to export disorder so efficiently that the overall disorder remains unchanged, that perfection is never achieved. Instead, living organisms export far more disorder than is required for them to maintain order in themselves. As a result, living organisms are net producers of disorder.
In that respect, people are much more vigorous producers of disorder than most other living organisms. People seek order not only in their bodies, but also in the objects around them and they achieve this ordering by consuming order in their environment—fossil fuels, minerals, pure water—at a furious pace and producing disorder in its place—burned gases, garbage, polluted water. Fortunately, sunlight is a tremendous source of order for our earth and it undoes some of the disordering caused by living organisms. However, we are consuming much of the order that sunlight stored on earth over millions of years in only a few generations. At this pace, we're destined to have troubles with the disorder we're creating. Many of the environmental issues that face us today can be viewed from this order/disorder perspective: we have to learn how to create less disorder.
Thermodynamics is a statistical science that deals with systems that are so complicated or vast that they can't be followed in complete detail. It makes predictions of behavior based on probability theory and while some of its laws predict probable outcomes rather than certain outcomes, a sufficiently probably event is effectively a certain event. For example, I can say with near certainty that if you play the lottery 50 times, you won't win the jackpot 50 times. I can't be truly certain of that fact, but the likelihood of my prediction being correct is pretty good.
In a sense, probability is destiny. Thermodynamics observes that vast systems tend to evolve toward the mostly likely configurations. To understand this process, consider what happens when you mix hot and cold water. The most likely final configuration for the mixed water is for it to reach a uniform temperature about half way in between the two original temperatures. While it's possible for the water to end up extremely hot in one place and extremely cold in another, that outcome is extremely unlikely. It's so unlikely that it never happens.
So in what sense does thermodynamics overwhelm things? The world is filled with relatively ordered arrangements and these ordered arrangements are unlikely by themselves (how they came to be ordered in the first place is another matter for another questions). If you take a crystal vase and drop it on the floor, it's going to evolve toward a more likely arrangement of atoms and dropping it a second time isn't going to return it toward its original unlikely state. In short, ordered systems naturally drift toward disorder when given a chance. How quickly they drift depends on their situation. A coffee cup will remain a nicely ordered object for thousands or millions of years if you don't disturb it. But in a hot environment, or one that is chemically aggressive, it may not last very long.
One last thought: how do living organisms maintain their order in the face of this tendency to disorder? They do it by consuming order and exporting disorder—they eat ordered foods and release disordered wastes to their surroundings.
An air conditioner uses a condensable working fluid—a chemical that easily converts from a gas to a liquid and vice versa—to transfer heat from the air inside of a home to the outside air. This process involves three major components and at least one fan. The three major components are a compressor, a condenser, and an evaporator. The compressor and condenser are usually located on the outside air portion of the air conditioner while the evaporator is located on the inside air portion. The working fluid passes through the insides of these three components in order, over and over again, so I'll start examining what happens to the working fluid as it enters the compressor.
The working fluid arrives at the compressor as a cool, low pressure gas. The compressor squeezes this working fluid, packing its molecules more tightly together so that their density and pressure increase. The squeezing process also does work on the working fluid, increasing its energy and therefore its temperature. The working fluid leaves the compressor as a hot, high-pressure gas and flows into the condenser. The condenser has metal fins all around it that assist the working fluid in transferring heat to the surrounding outdoor air. As this transfer takes place, the closely spaced molecules of the working fluid begin to stick to one another, releasing additional thermal energy into the surrounding air and causing the working fluid to transform into a liquid. By the time the working fluid leaves the condenser, its temperature has almost dropped back down to the outdoor temperature but it is now a liquid rather than a gas.
This high pressure liquid then flows into the evaporator through a narrow orifice. This orifice allows the liquid's pressure to drop so that it begins to evaporate into a gas. As it evaporates, it extracts heat from the air around the evaporator because that heat is needed to separate the molecules of the working fluid. Like the condenser, the evaporator has metal fins to assist it in exchanging thermal energy with the surrounding air. By the time the working fluid leaves the evaporator, it is a cool, low-pressure gas. It then returns to the compressor to begin its trip all over again.
Overall, the working fluid releases heat into the outside air and absorbs heat from the inside air. The direction of heat transfer, from a cooler region to a hotter region, is the reverse of normal and requires an input of ordered energy so that it doesn't violate the second law of thermodynamics (the disorder of an isolated system can never decrease). This ordered energy is used to operate the compressor and is converted into thermal energy in the process. This additional disordered thermal energy enters the outside air and makes up for the additional order that's given to the indoor air as that air is cooled.
I assume that you are referring to the gambling machines that spin several wheels when you pull a lever and that pay you amounts that depend on the patterns of symbols that show on the faces of the wheels when they stop. While the final arrangement of symbols that appear on such a machine when it stops is entirely random, the patterns that pay and the amounts they pay are calculated to ensure a slight financial advantage for the house. The mathematics of probability is well developed for such gambling machines and it's relatively simple to determine what fraction of your money you should expect to lose if you play the game for a very long time. If you do play long enough to sample the full statistics of the game, you are certain to lose money. It's only if you play briefly that you can take advantage of statistical fluctuations to leave with more money than you had when you started.
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