This atom is even more likely to create that quantum if other quanta of that same energy are already in the neighborhood! So if you have a gas of these atoms, and you illuminate it with a beam of 2.0 eV quanta, you can stimulate the atoms to lose energy and create new 2.0 eV quanta.
(The new quanta also have the same phase and direction as the old ones -- they're identical in every respect. We call this coherent light.)
2.
In 1923 Compton found that X-rays scattering off metal increase their wavelength by an amount that depends on the scattering angle, with the maximum increase occurring for scattering straight backwards towards the X-ray source. If we think of light as a wave then it makes no sense that changing its direction should increase its wavelength; after all, when you look at yourself in the mirror, the blue in your clothing looks blue, not green, and the red in your clothing looks red, not infrared! But if light can also be thought of as particles (quanta) then it makes perfect sense: a high-energy X-ray quantum is scattered by interacting with an electron (which typically has much lower energy) in the metal and giving some of its energy to that electron. Hence this quantum now has lower energy than it started with. The relationship E = hf says that reducing a quantum's energy amounts to lowering the corresponding wave's frequency, and lower frequency corresponds to longer wavelength.
The great significance of Compton's work was immediately recognized, which is why he was awarded the Nobel Prize in physics just four years later.
3.
The "de Broglie wavelength" of a particle is inversely proportional both to the particle's mass and to its speed. (A simpler way of saying this is that it's inversely proportional to "momentum," which is just the product of mass and velocity.) So long-wavelength "de Broglie waves" (or "matter waves") correspond to slow-moving, low-mass particles.
4.
As Bohr first hypothesized, the energy of the emitted light and hence its color (wavelength) is determined by the energy difference between the two energy levels: that's just conservation of energy, since the emitted light must carry away exactly as much energy as the electron has lost in its downward jump. This must be true whether the emission is stimulated or spontaneous.
But the direction of this emitted light is another matter. In spontaneous emission the direction is random -- as is the timing of the emission. (Einstein never liked this, and hoped eventually to restore cause and effect to the process by showing why the emitted quantum headed off in a particular direction and was emitted at a particular time.) Stimulated emission, on the other hand, always produces a quantum that travels in the same direction as the quantum that did the stimulating. (It also is in perfect synch with the stimulating quantum.)
5.
Once the results of the Compton scattering experiment became known in 1923, it seemed to many as if Einstein had been right after all in 1905: light consists of quanta. But this result was so counterintuitive -- given all the experimental results that indicated that light is a wave -- that Niels Bohr and his collaborators Kramers and Slater were willing to try a radical alternative. They claimed (in a somewhat vague way) that all of the electrons in the metal interact with each other in a way that causes them to gain and lose energy. Thus the scattered X-rays Compton observed were not directly related to (i.e., were not directly caused by) the incoming X-rays, and hence one didn't have to treat the incoming X-rays as a stream of particles (quanta) that "hit" the metal, lost energy, and scattered off in some new direction. This also predicted that recoiling electrons were not directly produced by the incoming X-rays.
Using a billiards analogy, imagine that the cue ball approaches the stationary eight-ball, the eight-ball flies off to the right a half-second before the cue-ball reaches it, and then 1.7 seconds later the cue ball slows down and veers leftward. The end result may conserve energy, but it wasn't conserved at all times: for example, energy was created from nothing when the eight-ball started moving even before the cue ball slowed down. Now make things even stranger by supposing that even the end result might involve some net creation or destruction of energy -- for example, that the eight-ball gains a bit more energy than the cue ball eventually loses -- but that all the balls on the table are doing this kind of thing, with energy being created in some instances and destroyed in others, such that the total energy of all the billiard balls stays roughly constant. This is the BKS idea: rather than having cause-and-effect processes, each of which conserves energy at every moment, we have random uncaused effects that only conserve energy on average.
6.
Neither choice is correct. Millikan, like almost all physicists, thought that Einstein's quantum hypothesis was absurd, and he set out to disprove it by obtaining precise data on the photoelectric effect. Einstein predicted that the only thing that would influence the stopping voltage was the frequency of the incident light, and that above the threshold frequency the graph of stopping voltage vs. frequency would be a straight line. Millikan, on the other hand, expected that the metal's temperature would be the most important factor determining the stopping voltage -- presumably because hot metal contains electrons that, on average, vibrate with high energy.
Millikan's careful experiments clearly showed that temperature had nothing to do with it and that Einstein's predictions were right on the money. Yet while Millikan was willing to dump his preconceptions with regard to temperature, he still insisted that Einstein was wrong about quanta. Even as late as 1923, in the lecture he gave while accepting the Nobel Prize in physics, he conceded that Einstein's equation was correct -- there was a linear relationship between frequency and stopping voltage, and nothing but frequency mattered -- but said that Einstein's underlying theory that had led him to this equation was wrong.
7.
If a moving baseball has wave properties, then a baseball crashing through a window is somewhat like a laser beam (a light wave) passing through a slit. So if the laser produces a diffraction pattern on a viewing screen, why doesn't the baseball produce a diffraction pattern on the wall opposite the window, a spread-out pattern of "baseball not-baseball baseball not-baseball baseball ..."?
To see the answer, ask yourself why a laser beam doesn't produce a diffraction pattern when it passes through a large opening like a window or an open doorway. We've seen that diffraction is important when an opening (or obstacle) is small. Small compared to what? Compared to the light's wavelength. Thus it's easy to observe diffraction of long-wavelength waves -- as when the bass notes from the stereo down the hall easily bend around the edges of your open doorway -- but you can't notice diffraction of short-wavelength laser light unless you use narrow slits or narrow hairs.
Macroscopic objects like baseballs have a lot of mass and hence a lot of momentum, so their de Broglie wavelengths are minuscule -- vastly smaller than the width of any conceivable opening or obstacle. That's why they never seem to behave in a wavelike fashion. Don't get mixed up and talk about the baseball's diameter as being smaller than this opening or larger than that opening. Diameter is a particle property, whereas de Broglie is asking us to think about the baseball as a rapidly oscillating pattern spread throughout the room rather than as a small sphere with a finite diameter.
8.
Bothe and Geiger used "coincidence counters" to compare the times at which scattered X-rays and recoiling electrons are produced in Compton scattering. Einstein saw it as a cause-and-effect process: an X-ray quantum "hits" (interacts with) an electron in the metal, gives some of its energy to the electron, and is thus deflected (scattered) in some new direction while at the same time the electron goes flying off (recoils) in yet another direction. It's essentially like a billiards shot, with the incoming particle (quantum) of X-ray light playing the role of the cue ball. Bohr, Kramers, and Slater, on the other hand, had hypothesized that the recoil of the electron was not directly caused by the incoming X-ray light and hence the electron might leave the metal before or after the X-rays scattered from the metal -- at the same time on average but only on average.
The results? Bothe and Geiger found that the two processes always coincided, always happened simultaneously. So BKS were wrong and Einstein was right: light quanta exist.
Compton himself carried out a separate follow-up experiment that measured the direction of the recoiling electron's motion. If it's really like a billiards shot, you can predict the angle at which the electron will recoil if you know the angle at which the X-ray quantum scatters. Again, this experiment showed that Einstein's quantum hypothesis was correct.