The first question has an answer that can be accomplished in QFT. In that context is shown that vacuum is not "empty", because is always filled by infinite processes which depend on the scale of observation. Moreover QFT-vacuum can have interesting condensate structures, also depending on the motion of the observer (see Unruh effect).

Hi,

Vacuum is defined as "space void of matter". See for example

https://en.wikipedia.org/wiki/Vacuum

Space is defined as the boundless three-dimensional extent in which objects and events have relative position and direction.

https://en.wikipedia.org/wiki/Space

A sealed "empty" box should be considered as a composition of elementary particles which move through the empty space. Of course, nothing happens with the space occupied by particles.

Hope this helps.

So, is the  "vacuum" embedded in "space"?

No, it is just space without matter.  On the other hand, in engineering and applied physics, vacuum refers to any space in which the pressure is lower than atmospheric pressure. Even if all matter could be removed from a volume, it would still not be "empty" due to vacuum fluctuations, dark energy, and other quantum phenomena. 

Does the space inside the box remains the same after the box is displaced or the box covers a new region of space after it has moved?

As I understand, it is in the space where these vacuum fluctuations occur, so the "space" provides the space for vacuum to exhibit its properties.

Does the space inside the box remains the same after the box is displaced or the box covers a new region of space after it has moved?

"Empty space is not empty" That's the wise word of the theoreticians.  But they try and give an intuitive picture where there is only obscure mathematical formalism. No, empty space is empty, no physical manifestation of empty space can show without using matter, then empty space is of course no longer empty.  Hawking radiations and Unruh effect are only purely theoretical constructs without experimental confirmation.  Renormalization is only a dubious mathematical procedure that has no identifiable physical counterpart.

For the second part of the question, (empty) space is not a substance, it can be displaced by a mere mental operation, which is called a symmetry, and is associated with conservation of momentum. In general relativity the view is a little different. Say there is a gravitational field in the box, that is, space is curved.  This curvature isn't displaced with the box, assuming it is light enough. But a gravitational field is much like a spatial relationship with heavy bodies.

Vacuum is not empty. I mean the vacuum of quantum field theory, i.e. the state of lowest energy, where quantum excitations (particles) are created because of the presence of quantum fields (which fullfills the spacetime). It can present, e.g., a rich condensate structure (see BCS model).

Luca, state of lowest energy and quantum excitation are contradictory. The zero point fluctuation (ZPF) is the lowest energy of a mode, the one with a half hbar omega. Any excitation is a quantum of one hbar omega. Virtual particle would appear because of Heisenberg's uncertainty relation, but that is quite different. The ZPF is constant, uniform, and infinite when summed over all the modes.

What does "empty space" mean at all? In our current understanding, fields pervade the whole spacetime. If empty means lacking a mass-energy density, well, this is only one side of the story. Without getting into the quantum realm there would be a metric defined, and a Curvature, and a Ricci Tensor. Even if, say, at point x the energy-pulse tensor is zero, that does not imply a zero Ricci Tensor; and what's more fields have proven to be more than a useful construct: they are real,physical entities(Aharonov-Bohm for example). Lack of matter doesn't mean empty as far as I am concerned. As for vacuum, it may have a noteworthy structure, and it's true that in general it does have such a structure. Exhibiting such symmetries or asymmetries the vacuum is again by no means "empty", since with this line of reasoning you should extend the word " empty" to the presence of excitations as well.

I didn't read the answer of Claude Pierre Massè before. If you read carefully my answer, you'll understand that I didn't say that vacuum is generally made by field excitations. I said that particles are field excitations and vacuum is the background of this scenario. However, the vacuum itself can be made by particle condensates (see BCS theory, Bogoliubov model of superfluids, flavor vacuum for neutrino mixing and many other examples of vacuum as generalized coherent states).

The vacuum is not empty. If it would be empty, it would have null energy, what is a fix number, and would conflict with Heisenbeg's principle. So it must have energy variation e.g. vacuum fluctuation. But according to Einstein's equation in the general relativity, the spacetime and the energy or the matter cannot exist separatelly, so vacuum cannot embedded into the empty spacetime.