What is the principle behind diffraction? If we shine a green laser on a wall in a dark environment, we would observe a bright dot in the middle. Now, if we fix a grating just in front of where the laser is held, when the beam strikes the grating, the photons would spread and form a grid of dots on the wall instead of one at the center.
The traditional explanation is that the grid of dots form because of the spread of light as the photons exit the aperture, where they start expanding and interfere with each other.
For example, in Why does a wave actually diffract?  and in Why does light diffract only through slits?, the general consensus seems to be that light diffracts all the time, not just through slits, but that this radial nature of expansion only becomes apparent once we add the slits. However, in that case, we would expect the laser to reach as far in every direction with or without a grating at the front.
Edit: Another explanation that uses Huygen's principle as suggestification is in Khan Academy. if you watch the video on Young's Law, then you would figure out that the often recounted analogy of why light expands when it enters a slit is that there is nothing special about the slit, but that the expansion is a general property of light, aka Huygen's principle. The problem is that it doesn't really explain why diffraction happens.
This view is inaccurate because the slit affects the overall angle of produced light. Otherwise, laser light should expand without any slit at least as much as if there was a slit (grating), because the principle of Huygen should apply equally whether or not there is a grating. But through my school experiments, I know that a slit will sharply increase the angle of expansion, which does not agree with the proposed explanation. The conceptual explanation lacks any explanation for why the total angle of diffraction would increase.
It appears that the real reason behind why a grating causes light to spread is that it hits the edges of the slit as @Solomon argues. The edges of the slit cause it to bend in other directions, causing a larger diffraction.
 A: I surmise the focus of your question is the following:
If it is assumed that Huygens' Principle in its 17th century form is applicable then any propagation of light should be diverging all over the place, and it would not be possible to have such a thing as a beam of light.
You phrased it as follows:

Huygens principle does not only apply once it hits the grating, but also before and after it hits


Part of the answer is:
Obviously the 17th century version of Huygens' principle is insufficient; there is more to it.
We observe the following:
Sunlight entering a room through a hole with a diameter many, many times the wavelength of light will tend to propagate as a beam. Still, with sufficiently accurate measurement it can be demonstrated that some divergence of the beam does occur.
The smaller the hole, the more divergence.
When the size of the aperture is in the range of the wavelength of the light there is almost complete divergence.
In overview:
The amount of divergence correlates with the ratio of amount of edge relative to the size of the aperture. (A diffraction grating maximizes that ratio.)
To my understanding:
Fresnel proposed a modification of Huygens' principle, this proposal is referred to as Huygens-Fresnel
To my understanding:
Fresnel's proposal involves ad hoc assumptions, rather than being a derivation from first principles.

I'm not sure there has ever been an attempt to formulate a rigorous theory of diffraction. There is an abundance of observational data; the models that are used are adjusted to fit the observations.


On the question of how to understand propagation of light:
In order to formulate a theory at all the phenomenon of particle/wave duality has to be accommodated.
While the interaction of light with a detector is particle-like, the propagation is wave-like.
One way to illustrate that: the experiments with the source of the light so strongly attenuated that the detector as a whole detects one photon at a time. It is tempting to believe that the light is propagating one photon at a time. However, in transit there is no lower limit to the amount of electromagnetic energy per unit of time. It is still necessary to think of the flux of energy as continuous. It is the interaction with the detector where the transfer of energy can only happen on a photon-by-photon basis. A lower flux of electromagnetic energy means less occurrances of photon detection per unit of time. The spacing in time of photons being detected is in accordance with random distribution.
Summerizing:
Propagation: continuous
Interaction with matter: one photon at a time.
A: You are right. The opening of the laser should cause diffraction. In fact as Why does a laser beam diverge say, it does.
The leading answer says diffraction is a quantum effect, originating in the Uncertainty Principle. This is true.
But you can get the same answer from classical physics. The Uncertainty Principle is tied up in the wave nature of a photon. Classical physics also describes light as a wave, a solution of a wave equation derived from Maxwell's equations. To a first approximation, light propagates in straight lines. Diffraction is the difference between that approximation and the true solution. Again, diffraction originates in the wave nature of light. The two explanations are equivalent.
The ideal laser cavity is one where a wave propagates back and forth between two mirrors many times with minimal loss. As you might expect from the uncertainty principle, light between to flat mirrors would spread out away from the axis and be lost. To counteract this, cavities have at least one spherical mirror to direct light toward the center. This produces the ideal laser beam, a Gaussian Beam. This is the solution to the wave equation with the boundary conditions that the curvature of the wave fronts matches the curvature of the mirrors during reflection.
This beam is a little counter intuitive. The rays (lines perpendicular to the wave fronts) are not straight. They follow hyperbolic paths that are usually very close to straight. The reason is that the intensity is not uniform. The beam is most intense in along the axis. It fades exponentially away from the axis. In theory it never reaches $0$, even at large distances from the axis. The drop in intensity is not as abrupt as it would be from passing through a pinhole. But the same physics applies. The beam spreads out.
A: One way in which the propagation of light is viewed says that, unless the light is an eigenmode of the system in which it propagates, it will diffract. In free-space, there are different ways to represent the eigenmodes. One way is to represent them as the plane waves. This choice naturally leads to a Fourier optics approach.
Any beam of light can be represented as a superposition of plane waves. The coefficient function of such a superposition is called the angular spectrum (representing the "angles" of the propagating plane waves). To see how a beam propagates, one would first compute the angular spectrum (by taking the Fourier transform of the beam profile) and then use it to reproduce the beam at some propagation distance. Since the plane waves remain plane waves when they propagate, the only effect is that the plane waves pick up different phases. So we just multiply the angular spectrum with this phase distribution and compute the inverse Fourier transform. As a result, this varying phase distributions causes the beam to diffraction during propagation. This approach is completely rigorous, since it does not require any approximation. Neither it require any quantum physics. It is a purely classical theory.
When we do this calculation for a Gaussian beam profile, we see that the beam spreads out as it propagates. The rate of spreading is given by a beam divergence angle, which is given by
$$ \theta = \frac{\lambda}{\pi w} , $$
where $\lambda$ is the wavelength of the light and $w$ is the width of the Gaussian profile. This angle it not so difficult to measure. Any laser beam has such a divergence angle.
What about the Huygens principle? Well it turns out that one can interchange the order of the two integrations associated with the Fourier transform and the inverse Fourier transform. In the process, the first integration then produces a kernel function with which we need to convolve the input profile function to obtain the output profile function. Unfortunately, this first integral is not tractable (it is the integration over all plane waves). However, one can perform it numerically to get an idea of what this kernel looks like. It turns out to be reminiscent of a sphering wave with an amplitude that decreases toward the sides. So in effect, the convolution process is implementing a Huygens principle.
Hope my explanation does not use too complicated language. If anything is not clear, let me know.
A: You are overlooking interference.
Light has a tendency to spread. Huygens' method is to assume light propagates by spreading in a semicircle (ie at all angles) from every point in space that the light reaches. The consequence of this spreading is that light from one point will interfere with light from another, and the outcome of the interference is what counts. So if light from very many points spreads in all directions, the interference can still have the effect of producing a narrow beam because waves heading at other angles interfere destructively and cancel.
The introduction of slits and gratings into the path of a beam of light reduces the scope for interference, since light from points in space that were previous contributing to the overall effects of interference is now blocked and prevented from interfering, which results in a different pattern of propagation.
It is not that the slits themselves change the angles of propagation of the light- the light for every point always does spread in all directions, and the slit does not change that. What the slit does do, is to prevent some of the light from propagating at all, and the absence of the blocked light allows th remaining light to spread further without being cancelled out by interference effects.
A: 
If we consider light as expanding in every direction by default, with every wavelet as a source of a secondary wave,

This is a wrong model for classical electromagnetic radiation. There are no "wavelets" of light. This is how mathematically electromagnetic radiation is fitted to the data. Only interacting with electric and magnetic fields of matter will the light ray be scattered.

Light rays do not interact with light rays, they go through each other so the "new source" for wavelets you imagine cannot exist in vacuum, there has to be matter on the way which will scatter off the ray but this, for laser beams , has a low probability so the beam goes through undisturbed. This answer of mine is relevant.
