On dark matter distribution and effects at terrestrial (or solar system) scale Or the lack of them. If DM is much more abundant of common baryonic matter, why its gravitational effect becomes evident only if the dynamic of huge systems such as galactic rotation is considered?
I do not understand why DM should not contribute to our weight (assuming that it permeates Earth or it is concentrated in its nucleus) or should not be considered when describing the motion of the solar system.
What I do not understand here? I am ok with all instances that brought DM into consideration but I do not see why its gravitational effects do not take places here on Earth . Shall I take it as homogeneously distributed (at the scales I mentioned) so that I and the Earth are basically "within a DM spherical shell" and, thus, basically facing each others as if DM was absent? 
Edit. Likely the moderators will add this. My question is about a duplicate of
Why doesn't dark matter affect planetary motion? 
 A: Because there is little dark matter in the vicinity of the solar system. You might've seen plots of galaxy rotation curves, which tend to look like this:

Note that near the galaxy center, the predicted orbital velocity matches the observed one almost exactly. In other words, there is very little dark matter near the galaxy center; almost all the dark matter is in the periphery of the galaxy. The Solar System is not that far from the galaxy center, and is in the region where ordinary matter dominates.
Now why did galaxies form such that ordinary matter ended up near the center while dark matter dominated the outskirts ... that is another question entirely.
A: It is analogous to why the center of the earth is not full of air.  
Theoretically dark matter is called so because it is postulated that it is attracted only by gravity, and thus by galaxies and clusters of galaxies. 
When the solar system formed 

The formation and evolution of the Solar System began 4.6 billion years ago with the gravitational collapse of a small part of a giant molecular cloud. Most of the collapsing mass collected in the center, forming the Sun, while the rest flattened into a protoplanetary disk out of which the planets, moons, asteroids, and other small Solar System bodies formed.

Note the molecular cloud, these are normal matter which when attracted by the gravitational field formed  denser and denser masses, the gasses always on the outside of the balls due to the other three forces. Dark matter is postulated to have only the gravitational interaction and no bound states can form.
A dark matter particle will be attracted by the earth but it will not be stopped by ordinary matter, like neutrinos, it will go through to the other side, or be caught in an oscillation if its energy is low.
It is as a type of gas and the earth will be trailing the amount that corresponds to its mass, but it will be spread out in space, the same with the solar system. It is only at the sizes of galaxies that the gravitational effect of so much extra matter becomes evident.
There is this interesting proposal of simulations with dark matter in the solar system here .

I do not understand why DM should not contribute to our weight (assuming that it permeates Earth 

Take the earth as an example, and an imaginary tunnel to the center. The gravitational attraction, 1/r^2 acts from the location r to the center r=0, The masses above r do not contribute. As the gas of dark matter is extremely diffuse only a small part will be below the earth's surface and its effect on the weight will be infinitesimal. 

or it is concentrated in its nucleus)

It is not concentrated at the center, as discussed above.

or should not be considered when describing the motion of the solar system.

As for the solar system:

The average density of dark matter near the solar system is approximately 1 proton-mass for every 3 cubic centimeters, which is roughly 6x10-28 kg/cm3. The actual density might be a little lower or higher, but this is the right order of magnitude.

It is comparable to the interplanetary medium which is not used in calculating planetary orbits because it is so weak.
The research paper in the fourth link above does consider the effect of the gravitational poles, like the earth, on dark matter in a model built specifically for this study. If the model holds, and dark matter consists of weakly interacting particles the calculations may be useful for positioning detectors to detect their very weak interactions.
