Can space be distorted without things that occupy the space being distorted? I mean it is whether in reality or imagination?
The answer depends on whether you are interested in this topic from a philosophical point of view, or a physics point of view.
In the case of the philosophical point of view, there is no right or wrong answer. Anyone is entitled to argue the point any way they wish, and you are free to select whichever argument you prefer.
From the physics point of view, there is a right answer, provided to us by Einstein: massive objects embedded in curved space will experience the effects we associate with gravity. Matter tells space how to curve, and curved space tells matter how to move.
Yes, this is what happens. The universe is expanding, but because matter is overwhelmingly clumped into galaxies, they don't feel the effects. Ultimately this is now expected to lead to a 'big rip', where clumps of matter become completely separated from each other by cosmological event horizons. These cosmic effects depend on whether space is closed, flat, or open/saddle-shaped, which depends on density and concentration of massive particles. The universe began we think with only massless energy. But no one knows yet why the expansion is accelerating, so clearly we don't have a full picture. Other forces are gigantically more powerful than gravity, that's why you can jump in the air, a few electrons resisting the gravity of the entire Earth.
There is thought to be a 'gravitational wave background', at long wavelengths and low energies. Our current sensing can only pick up the highest energies and shortest wavelengths. I have heard estimates that our solar system is vibrating by around 300 feet, due to this background.
According to Leibniz's Principle of sufficient reason, space cannot be thought of as some absolute container or background platform in which all objects are created and evolved:
Leibniz also used the principle of sufficient reason to refute the idea of absolute space: I say then, that if space is an absolute being, there would be something for which it would be impossible there should be a sufficient reason. Which is against my axiom. And I prove it thus. Space is something absolutely uniform; and without the things placed in it, one point in space does not absolutely differ in any respect whatsoever from another point of space. Now from hence it follows, (supposing space to be something in itself, beside the order of bodies among themselves,) that 'tis impossible that there should be a reason why God, preserving the same situation of bodies among themselves, should have placed them in space after one particular manner, and not otherwise; why everything was not placed the quite contrary way, for instance, by changing East into West.
Thus Leibniz rejects Newton’s space absolutism and substantivalism which assume space and time are entities in their own right, existing independently of things occupy it. Leibniz’s space relationism, on the other hand, describes space as a system of relations that exist between objects. Under this view, space is absorbed as a mere relational property of an object, so there can be no such possibility that "space be distorted without things that occupy the space being distorted".
Now even if you subscribe to space substantivalism as assumed in modern Einstein's general relativity conceiving ambient space as substantial manifold (classically regarded as 3-d), it's still hard to imagine the possibility of space be distorted without things that occupy the space being distorted. This is just like our common view, yet how can you be sure we're not distorted already since our ambient space may be a non-Euclidean world? Normal small objects are just localized in space and thus cannot feel any distortion of itself, not to say ambient space. Imagine there exist 2-d human beings on a 2-d sphere, their body shapes have to conform to the shape of their containing space, thus the sum of angles of a triangle in this spherical world would be distorted and greater than 180 degrees relative to our supposed flat world, but for the 2-d localized human being they cannot feel this distortion at all. It's similar on earth, if we're big enough we can immediately see the earth is not flat.
In summary, if your proposition "distortion of object and its ambient space can be separated" is true in some world, then from our experience or Kant's synthetic a priori theory of space, it's hard to see any meaningful relation can exist between object and its occupying space, seems no epistemic sense to even link them...
It depends on the object. If the object is constituted of connected parts that have a fixed distance wrt each other, then upon twisting and stretching the space these distances will not become smaller or bigger. The object will stay the same.
To be explicit, consider a flat two-dimensional space. There is an object in it consisting of a collection of particles that form a square grid. You can say the object is a solid square. The distances of the particles wrt each other are fixed. If you distort this space, the particles will not move away from each other. Only if they were attached to this space, this would happen. So there will no change be visible in the object other than that it will follow the curvature of the space which is not visible from within this space. If you stretch the space only, the square will not change at all. So when the square is looked at from the inside, no change will ever occur, except when the particles will be pulled apart, but considering the assumption that their mutual distances can't vary, this can't be done.
If the object is made up of loosely connected parts and the distance of these parts is variable then the object will change in form because the distances between them change. So twisting as well as stretching will affect the form. The twisting effect can't be seen from the inside though. The distances between the part(icle)s will stay the same if the space is only twisted. Only on the outside, the effect will be visible. When stretched space will induce variations in the distances so seen from the inside the square can change in the form yo want it to.
No, space cannot distort (that is: absolute space, as presented to us, and not the abstract mathematical construction of space). Just try to imagine it distorting, I can't. Even mathematical abstractions like curved space are always visualized by a projection. I suggest you look into the topic yourself, but I'll give you a start below.
The first two attempts at understanding space that I'm aware of come from Newton, and Leibniz, who approach it naively as something "out there, in the world". Newton understood it as a stationary entity [https://plato.stanford.edu/entries/newton-stm/]. This was easily refuted when it was pointed out that it would be impossible to tell whether it was stationary. E.g., is the bucket rotating, or is space? Leibniz understood it as an effect caused by the distance vectors between objects, a theory which is still popular today in speculative physics because it yields to mathematical analysis making it a fun toy (but not real, in any meaningful sense) [https://www.researchgate.net/publication/257563246_Leibniz%27s_Theory_of_Space].
The problem with thinking of space as "out there, in the world" is that it is like nothing else that we think of that way. For one, there are infinitely many spaces for objects to occupy in space, and they are all identical to each other except by the fiat of imagining objects which constitute their boundaries. Second, it's is continuous, whereas everything else "out there, in the world" is individuated insofar as it can be imagined at all. And lastly, for our purposes, there is no mental image without space, and there is no mental image of space alone (no, space is not empty blackness).