Do empty space in our universe exist or it is all filled with dark matter or some other unknown matter?
“Vacuum” is often loosely defined as the absence of matter.
Interstellar and intergalactic space has a very low density of atoms, so it can be considered a vacuum to a good approximation. However, every cubic centimeter of the universe actually contains about 400 photons left over from the Big Bang, so radiation fills the universe. There are also believed to be lots of leftover neutrinos everywhere, a kind of lightweight matter. Physicists also think dark energy (not dark matter) is everywhere. Dark matter is believed to mainly clump around galaxies, so there is probably not much of it between galaxies.
An ideal vacuum would have no matter (including no atoms, no neutrinos, and no dark matter) and no radiation (such as photons) either. But it could still have dark energy, which is sometimes called “vacuum energy”.
Physicists think of the ideal vacuum state as the state in which there are no field quanta but there can nevertheless be field energy.
From the perspective of quantum field cosmology, G Smith has answered the question comprehensively. However, a simpler answer might be that at a more fundamental level than currently admitted by physics, all of reality, matter and space, even the very mind, is made up of the same thing, a singular substance, whose dynamic behaviour translates to what we loosely call 'energy'. Unfortunately, not only does this sound alarmingly like philosophy, generally reviled by mathematical physicists of the present day, but those physicists would baulk at the idea of energy as equivalent to the activity of a substance.
Yet when such a substance or fabric is understood to comprise the universal entirety, that idea becomes sound. For example, pursuing the point you quite correctly make yourself, that it is quite impossible for something to imagine nothing, and no doubt vice versa, were one to suppose similarly that the universe is composed entirely and exclusively of a singular inviolate substance, it would be quite impossible for us, also comprised of it, to determine its ultimate nature.
Further, that entirety, as you correctly imagine, must by very definition hold itself together; so that with this self-evident knowledge, a priori, one is required to inquire how it is able to do this without collapsing upon itself; to which the answer is that, since this 'cohesive effect' is universal and must operate at or through every conceivable spatial point, that force of 'cohesion' must be acting in disparate components at any and all such points; the holding together one way tends to balance approximately the holding together in the opposite direction at any point. When imagined with respect to a common archetype--say the purely geometrically-defined interval on a universal cubic lattice--, these components of a 'cohesive force' are effectively vibratory in origin and always slightly out of phase with each other within a range defined by half that interval, and any 'field theory' attempting to describe these vectors of cohesive force and their relations is of necessity bound fundamentally by constraints of such a 'phase relation'.
What results from this interaction between points defined by disparate resultants in such a 'cohesive force' then is a ceaseless motion as these components (arising eventually from the interior dimension of 'spatial depth' itself) act one against the other throughout universal reality in an inherent tendency to equilibrium (impossible of course--but don't tell them that)--or 'cohesive symmetry'. That motion we call energy, whether in the relative vacuum of space or in the midst of what we describe as 'matter' by virtue of our similitude with it, and consequently our capacity to perceive resonance with it (within a definitive range); and the distribution of that 'cohesive force', of which all ostensible forces are aspects, constitutes the properties of reality, mass, charge, momentum, and energy when that is understood as an oscillatory principle in that distribution. The reciprocal cubic lattice model of 'cohesive loci' mentioned is capable of representing these basic quantities for the electron and photon for example as functions of linear and planar area distribution of such cohesive force, but that is a story for another day (of a 'wave theory of universal resonance').
Science, and physics in particular divorced itself from an age-old foundation in philosophy long ago; it is probably high time they both rediscovered the simple joys of marriage.