Moving through curved space objects with volume OK, the topic of my post is General Relativity, and first I try to present my train of thought.
GR says that gravity is geometry, and what we may experience as a gravity force is in fact objects moving in curved geometry and following straight lines in that geometry.
I imagine a curved geometry like this: let's take a checkered paper. In euclidean geometry every square on such paper has got the same size, so if we draw two parallel lines, then they will never meet. But if we had a similar paper for a (positive) curved geometry then as get away from a central square, the squares gets smaller and smaller. On such a paper someone drawing two parallel lines will discover they cross each other in two points.
And the fact that we observe in our reality two objects getting closer and closer due to "the force of gravity" is because those objects move in time and because of the positive curvature of  spacetime.

So far so good, but all that treats objects as points. But what about objects with non-negligible volume?
If I understand right, if we have got a non-infinitely small object moving forward in curved space, it will experience an imaginary "pressure" from sides perpendicular to the move direction. In other words, that object should get squished. In 4D spacetime time is perpendicular to all space dimensions, so if we still treat gravity as geometry then it means objects should get smaller as they move through time.
But I have never seen someone taking that point when explaining GR. Moreover, when someone talks about objects with strong gravity (eg. black holes) then an opposite situation takes place: other objects approaching it get stretched (the famous spaghettification effect).
What I'm asking then is I don't see that happen. Yes, this explains why an object with volume feels gravity on itself, but it also gives such strange things like, for example, telling the Moon is squeezed not because of its own gravity but because it is in the Earth's gravity.
Do I miss something?
 A: 
In 4D spacetime time is perpendicular to all space dimensions, so if we still treat gravity as geometry then it means objects should get smaller as they move through time.


But I have never seen someone taking that point when explaining GR.

What you are describing here is the Ricci curvature. Probably you have seen discussions and just not recognized the meaning of the term. The Ricci curvature describes the change in the volume of a family of initially comoving geodesics. Here it is described in terms of coffee grounds:
https://math.ucr.edu/home/baez/gr/ricci.weyl.html

Moreover, when someone talks about objects with strong gravity (eg. black holes) then an opposite situation takes place: other objects approaching it get stretched (the famous spaghettification effect).

This is described by the Weyl tensor. The Weyl tensor is the companion to the Ricci tensor. It describes how an initially spherical collection of coffee grounds gets distorted into an ellipsoid. In extreme cases this distortion, stretching and squeezing, is the spaghettiffication. Note that the volume of the coffee grounds does not change, just the shape.

it also gives such strange things like, for example, telling the Moon is squeezed not because of its own gravity but because it is in the Earth's gravity.

Ricci curvature requires mass (stress energy) to be inside the sphere of coffee grounds. The Weyl curvature can still be present when the coffee grounds cover a region of vacuum. The spacetime around the whole moon would include both types of curvature. A cloud of coffee grounds completely around the moon would both decrease in volume (Ricci) due to the moon’s gravitation and also distort into an ellipse (Weyl) due to the earth’s gravitation.
Note that the reason that you hear about spaghettification in the context of black holes is precisely because we are dealing with Weyl curvature in a region of vacuum outside the black hole. That does not mean that Ricci curvature is not part of general relativity, just that it is not relevant for that specific scenario.
