How can a grain of sand be "spaghettified" when nearing a black hole? I have a hard time wrapping my head around this "spaghettification" process that apparently takes places when getting close to a black hole.
Gravity is proportional to the distance of the object exerting gravity. Everything in space is unimaginably HUGE. This applies to, e.g., both distances and size of different bodies. Earth is big, but our sun could contain more than one million Earths in it. And a grain of sand is negligible compared to Earth.
The difference in gravity between the "front" and the "rear" of a grain of sand sent towards a black hole should/must be negligible, considering how weak gravity as a force is, compared to e.g., magnetism. So how can this weak force and miniscule gravity gradient cause spaghettification on something as small as a grain of sand?
(I believe I can understand why something as big as a star would become spaghetti when closing in towards a black hole.)
Second question, if grain of sand would spaghettify in these circumstances, would something much smaller like a hydrogen atom also spaghettify? A water molecule? A free electron? Neutrinos?
 A: To answer the "elementary particle" tag of the question which is:

Second question, if grain of sand would spaghettify in these circumstances, would something much smaller like a hydrogen atom also spaghettify? A water molecule? A free electron? Neutrinos?

A water molecule will break apart into  electrons protons and neutrons, the hydrogen atom into an electron and a proton.
As Andrew says," we cannot do any experiments" near a black hole, so we have to rely on observations and mathematical models.
Our current model for hadrons , neutrons and protons, is that they are held together with the strong force which becomes infinite if the quarks are distanced, so in the current models, which do not have quantized gravity, I presume that protons remain whole while neutrons decay on the way to the singularity, into a proton a neutrino and an electron. Electrons and neutrinos are non decaying  elementary particles, , and as such, retain their identity in the present models.
A: Spagghetification occurs when the gravitational potential energy on the side of the grain of sand closer to the center of the black hole is much larger than the potential energy on the other side. The gradient in potential energy across the grain of sand leads to a force, and if this force is large enough it will be larger than the forces holding the grain of sand together.
For simplicity, let's assume a non-spinning black hole of mass $M$.
The potential energy on the "near side" of the grain of sand, at a distance $r$ from the black hole singularity, is
$$
U_1 = \frac{GM}{r}
$$
The potential energy on the "far side" is
$$
U_2 = \frac{GM}{r + d}
$$
where $d$ is the diameter of the grain of sand.
Now when $r \gg d$, it is true that $U_1 - U_2 \approx 0$. This is where your intuition likely comes in.
However, if we are imagining a grain of sand falling into a black hole and hitting the singularity, then we are ultimately imagining $r$ going to zero. So we cannot assume $r \gg d$.
In fact, we have
$$
U_1 - U_2 = \frac{GM}{r}\left(1 - \frac{r}{r+d}\right) = \frac{GM}{r}\left(\frac{d}{r+d}\right)
$$
As $r\rightarrow 0$, this energy difference will become arbitrarily large. So as $r$ becomes small enough, inevitably the tidal force ripping apart the grain of sand will become larger than whatever (electromagnetic) forces hold it together.
We expect spaghettification to happen to any finite size particle -- sand grains, hydrogen, neutrons, etc.
There are a few caveats, because most physicists don't expect general relativity to be a complete theory that tells us what happens near the singularity of a black hole. Probably we need a quantum theory of gravity to really answer these questions.
When you start to get to distances $r$ of order a Planck length from the singularity, most physicists would agree that quantum gravity will become important, so the predictions of GR are no longer a good guide. However, there are also some speculative ideas that even when $r$ is of order the horizon of the black hole, general relativity breaks down. So, a final word of warning may be that since we cannot do any experiments to really see what happens to grains of sand in a black hole, we should be humble about saying exactly what will or won't happen.
A: Spaghettification does not always happen outside the event horizon. You could fall through the event horizon of a supermassive black hole, like the one at the center of our galaxy, without suffering any immediate harm. But soon enough (quite soon, in fact) you would get close enough to the singularity to be spaghettified. And so would a grain of sand or a molecule.
A: 
The difference in gravity between the "front" and the "rear" of a grain of sand sent towards a black hole should/must be negligible, considering how weak gravity as a force is, compared to e.g., magnetism. So how can this weak force and miniscule gravity gradient cause spaghettification on something as small as a grain of sand?

That depends entirely on the size of the black hole. Supermassive black holes are indeed so large (Schwarzschild radius grows linearly with mass!), and by consequence the gravity differential so small that you would indeed be able to fall into it crossing the event horizon fully conscious. Sadly, you won't be able to telephone home to tell us about your experience.
The Schwarzschild radius of stellar objects are in the range of tens to thousands of kilometers. The corresponding gravity differential might tear an astronaut to pieces, but would certainly leave a grain of sand intact until it crosses the horizon.
However, if you had a black hole with the weight of the moon, the Schwarzschild radius is on the order of millimeters, and your grain of sand won't stand a chance to reach it intact.
Now, "spaghettification" is a bit of a misnomer. It only means that the forces that hold your grain of sand together are not sufficient anymore for the grain to retain its shape. It will disintegrate in the same way as if you tear it apart by other means. For that grain of sand, that means it will break into smaller particles like stones do. A steel bearing ball would indeed be turned into a small wire because steel does deform before it breaks apart. But brittle materials won't get turned into spaghetti at all, they will be ripped into a stream of dust particles instead.
A: The difference in gravity between the "front" and the "rear"
This is exactly the point. In addition, the scribble - like everything else, has some initial speed and direction - and therefore inertia; and, having fallen under the influence of the gravity of a black hole, this, albeit a meager inertia, is the reason that multidirectional forces act on different parts of this body, which destroy it.
A: A (pseudo) singularity could well be dimensionless and spaghetti could not be described, if within network, then not a singularity, which seems to be the case, also this would be dimensionless; as per canonical theory mass would be indistinguishable within it, no mass no dimension, other way to see it. There is an instance aside string theory that may apply, but it is very remote from the context of the question.
Probably, the energy transitional state of a grain of sand attracted by the "singularity" or the sand grain's wakefield is likely to be spaghetti similar.
