How does quantization arise in quantum mechanics? BACKGROUND
I'm trying to build an intuition for what quantization really means and came up with the following two possible "visualizations":


*

*The quantization of the energy levels of a harmonic oscillator is the result of a wave function that is confined in a potential well (namely of quadratic profile). It is the boundary conditions of that well that give rise to standing waves with a discrete number of nodes---hence the quantization.

*Photons are wave packets, i.e., localized excitations of the electromagnetic field that happen to be traveling at the speed of light.
On the one hand, #1 explains quantization as the result of the boundary conditions, and on the other hand #2 explains it as the localization of an excitation. Both pictures are perfectly understandable from classical wave mechanics and yet we don't think of classical mechanics as quantized.
QUESTION
With the above in mind, what is intrinsically quantized about quantum mechanics? Are my "intuitions" #1 and #2 above contradictory? If not, how are they related?
PS: Regarding #2, a corollary question is: If photons are wave packets of the EM field, how does one explain the fact that a plane, monochromatic wave pervading all of space, is made up of discrete, localized excitations?
My question is somewhat distinct from this one in that I'd rather not invoke the Schrödinger equation nor resort to any postulates, but basically build on the two intuitions presented above.
 A: Actually in the case of your #2 there is no quantization since the energy spectrum of plane waves is continuous: there is a continuous range of $k$-vectors and thus a continuous range of energies.  The wave packet is just a superposition of plane waves, with continuously varying $k$ (or $\omega$) so not quantized.
To highlight the difference I will refer to an old paper of Sir Neville Mott, "On teaching quantum phenomena." Contemporary Physics 5.6 (1964): 401-418:

The student may ask, why is the movement of electrons within the atom quantized, whereas as soon as an electron is knocked out the kinetic energy can have any value, just as the translational energy of a gas molecule can?  The answer to this is that quantization applies to any movement of particles within a confined space, or any periodic motion, but not to unconfined motion such as that of an electron moving in free space or deflected by a magnetic field.

A: I am trying to answer your title question, because your questions about the atomic absorption/emission being quantized, is covered in the other answers.
You are asking "How does quantization arise in quantum mechanics?", and "If photon are wave packets of the EM field, how does one explain the fact that a plane, monochromatic wave pervading all of space, is made up of discrete, localized excitations?". 
If you accept that our universe is fundamentally quantum mechanical, then you need to describe the forces that govern it, and you need to describe how the forces act on matter by propagating mediators.
The EM force needs to be quantized to fully describe its interaction with matter. Photons, quanta of light are the only way to describe how light interacts with matter at the level of individual absorptions/emissions.
The weak force is bound by the heavy mediators, the W and Z, and the strong force is bound by confinement, using gluons. Both are in this way fully quantized, when we describe how they act on matter.

In other words, the weak and the strong force are, in some sense, "fully quantum" in that their importance to our world comes completely from their quantized description

Are there weak force waves?
The only exception is gravity, where we do not yet have a full quantum description of how exactly gravity acts on matter by propagating mediators, the hypothetical gravitons. But as you say, the need arises, because we are trying to describe the universe in cases where the gravitational forces are extreme, and dominate over all other forces (singularity).
So the answer to your question is, you can beautifully describe the universe by classical theories, like EM waves and GR waves, if you want to go with big scales, but as soon as you are trying to describe how forces act on matter (exceptions are photon-photon or gluon-gluon interactions) on quantum scale (elementary particles) you need a quantized force.
A: Quantisation means that the classical description of a particle having independent position and momentum at any time is replaced by a probabilistic description in which these numerical properties are not fundamental to the description of matter, but are determined in measurement processes. As summarised by Paul Dirac:

“In the general case we cannot speak of an observable having a value
  for a particular state, but we can … speak of the probability of its
  having a specified value for the state, meaning the probability of
  this specified value being obtained when one makes a measurement of
  the observable.”

The difference between this and classical probability theory is that classical probabilities are determined by unknowns, but quantum probabilities are actually indeterminate. Mathematically a probability density can be split into a function and its complex conjugate using the Born rule (this much is trivial) and quantum superposition is then the natural way to describe a logical disjunction (the result of a measurement may be one thing $\mathrm{OR}$ another). This much gives us the structure of a Hilbert space.
It is not trivial, but it can be proven, that preserving the probability interpretation under time evolution requires unitarity, and that the conditions for Stone's theorem are obeyed. The general form of the Schrodinger equation follows. 
This much has been well established in mathematical foundations of quantum mechanics, but it is generally not covered in text books which are concerned with application, not foundations and interpretation. I have written a paper for the purpose of clarifying The Hilbert space of conditional clauses.
A: I would suggest you give more credence to your idea #1. Photons can be explained in the framework of idea #1. After all, the reason we need photons in our theory is to explain why light energy only seems to come in discrete units.
Here's a way of understanding the energy eigenstates of quantum systems that matches pretty well with your idea #1.


*

*Consider a the classical mechanics of some system, like a particle in a potential well. 

*Compute the period of the system as a function of the total energy of the system, $T(E)$.

*Energy eigenstates are those states where the wavefunction reinforces itself constructively as it propagates. Combining this idea with the Planck-Einstein equation we see that the the allowed energies are the ones that satisfy $E T(E) = 2\pi \hbar n$ for some integer $n$. Different systems have different $T(E)$ and solving this equation for $E$ in terms of $n$ yields the energy spectrum.


This system works heuristically for one-dimensional one-particle systems. It misses things like zero-point energy and gets constant factors wrong, and it is hairy to extend to more dimensions and particles, but it does tend to give you the right asymptotic structure so I think it's helpful conceptually. I suggest that you can use it the explain photons as well. 
Explaining the energy quantization of one-particle systems
In the one-particle world the classical state of the system is determined by a single position function. Given a classical trajectory $x(t)$ with total energy $E$ you look for the period of $x(t)$ so that $x(t+T(E)) = x(t)$. 
Note that I haven't mentioned boundary conditions. Boundary conditions are important in this idea insofar as they are what create periodic classical trajectories! Classical systems without attractive potential wells don't have periodic classical trajectories and so their quantum analogues don't have discrete spectra, just a continuous free spectrum. The physical idea is periodic classical trajectories, which can be caused by attractive potential wells, which manifest mathematically in boundary conditions in Schrodinger's equation.
Explaining the energy quantization of the EM field
In the electromagnetic world the classical state of the system is determined by an electromagnetic field function, $A_\mu (\vec{x},t)$. For a classical field solution $A_\mu(\vec{x},t)$ with total classical energy $E$ you look for the period $A_\mu(\vec{x},t+T(E)) = A_\mu(\vec{x},t)$ and then solve $ET(E) = 2\pi\hbar n$. If you do this you find that there are infinitely many solutions for $n =1$ corresponding to $E = \hbar c|\vec{k}|$ where $\vec{k}$ is some vector. For higher $n$ you find more solutions, $E = \hbar c n|\vec{k}|$. This suggests that the energy eigenstates of the quantum electromagnetic field come in particle like chunks where one particle has energy proportional to its momentum and you can have arbitrary numbers of these particles. These are what we call photons.
Note also that we have a quantized energy spectrum without any special boundary conditions to speak of. Again, the quantization comes from the periodic classical field solutions. In the case of the EM field the periodic field solutions arise because of the EM wave equation, rather than being caused by an external potential.
Now, there are a ton of problems with every step of this conceptual approach. For one, if do the math you immediately find that a classical field that is periodic in time (and hence in space?) doesn't have finite total energy!
However, I am arguing that your idea #1 explains photons, and so you should take idea #1 and the fundamental idea that explains both photons and the quantization of energy levels in simpler systems.
A: First and second quantization
Quantization is a misleading term, since it implies discreteness (e.g., of the energy levels), which is not always the case. In practice (first) quantization refers to describing particles as waves, which in principle allows for discrete spectra, when boundary conditions are present. 
The electromagnetic waves behave in a similar fashion, exhibiting discrete spectra in resonators. Thus, technically, quantization of the electromagnetic field corresponds to second quantization of particles.
Second quantization arises when dealing with many-particle systems, when the focus is not anymore on the wave nature of the states, but on the number of particles in each state. The discreteness (of particles) is inherent in this approach. For the electromagnetic field this corresponds to the first quantization, and the filling particles, whose number is counted, are referred to as photons. Thus, photon is not really a particle, but an elementary excitation of electromagnetic field. Associating  a photon with a wave packet is misleading, although it appeals to intuition. (One could argue however that physically observed photons are always wave packets, since to have truly well-defined energy they would have to exist for infinite time, which is not possible.)
This logic of quantization is applied to other wave-like fields, such as wave excitations in crystals: phonons (sound), magnons, etc. One speaks sometimes even about diffusons - quantized excitation of a field described by the duffusion equation.
Uncertainty relation
An alternative way to look at quantization is from the point of view of the Heisenberg uncertainty relation. One switches from classical to quantum theory by demanding that canonically conjugate variables cannot be measured simultaneously (e.g., position and momentum, $x,p$ can be simultaneously measured in classical mechanics, but not in quantum mechanics). Mathematically this means that the corresponding operators do not commute:
$$ [\hat{x}, \hat{p}_x]_- = \imath\hbar \Rightarrow \Delta x\Delta p_x \geq \frac{\hbar}{2}.$$ 
The discreteness of spectra then shows up as discrete eigenvalues of the operators.
This procedure can be applied to anything - particles or fields - as long as we can formulate it in terms of Hamiltonian mechanics and identify effective position and momenta, on which we then impose the non-commutativity. E.g., for electromagnetic field, one demands the non-commutativity of the electric and the magnetic fields at a given point.
A: Although Schrödinger titled his 1926 pioneering 4 communications 'Quantization as eigenvalue problem', this is misleading. The discretization by boundary conditions applies to classical waves in strings and resonators. Not the energy gets discrete values, but the wavelength and subsequently the frequency.
The quantization of the electromagnetic field to photons of energy hf has nothing to do with boundary conditions, too.
The stationary Schrödinger equation for the harmonic oscillator has the following mathematical property. Any given solution with energy, E, is connected with all other solutions of energies 
E+hf, E+2hf,...   and E-hf, E-2hf,...
This holds true for all solutions, not only for Schrödinger*s eigensolutions! This means, that this equation has got an intrinsic discrete structure, independent of any boundary condition.
Now, all sulutions except Schrödinger's eigensolutions represent perpetua mobilia and hence violate the energy conservation law. This makes them to be unphysical.
I agree that it is easier to visualize boundary conditions than recursion formulae.
More details are in publications by Dieter Suisky, who had the core idea, and myself under the title 'quantization as selection problem'.
Have fun! Peter
A: Answering just the title question I would say that $E=h\nu$ is the basic equation. It implies that matter, photons, electrons etc., is described by waves with a frequency and hence a wavenumber. It also implies that these waves are normalized to represent these energy quanta. Wave equations such as Schrödinger, Klein-Gordon and Dirac describe basically the Einstein relation between energy and momentum, $E^2=m^2+p^2$. The Noether theorem provides the expressions for energy and others conserved quantities. These observations can build the intuitive picture that you are looking for. 
