Discrete spacetime: what does it mean for spacetime fields and vacuum? Suppose we imagine that the space-time continuum is really a discrete stuff of something. This something could be some kind of particles or "substance"/matter/energy field. Then, I see two conceptual problems if we keep the broad picture of Nature made of quantum fields on a space-time substratum:


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*What is "between" two atom of space-time or atoms of space (and time) if different? Vacuum? Can distances or neighbourhood be defined if no space and no time and no field is defined?

*Vacuum is generally believed to be the main lowest state of field theory, on space-time. Supposing there is no space-time, can we even define what vacuum is? Does vacuum exist without reference to a particular background independent theory?
In summary, is field theory (including classical gravity) and the notion of vacuum into trouble if we assume that the spacetime continuum is a discrete set of "something"? What is vacuum in a discrete theory of spacetime?
 A: You should really look into Loop Quantum Gravity for a quantitative example. While unconfirmed and highly speculative, it does offer a toy example for a background independent quantum field theory, that is, a theory that describes the quantization of space-time rather than living on classical space-time.
I'll try to answer your questions 1 and 2 from the point of view of Loop Quantum Gravity. But first, I want to clear one very important conceptual point that is almost always misunderstood.
According to LQG, space-time isn't exactly discrete, nor is it continuous. Instead it is quantum. Quantum objects have been known to consistently combine continuous and discrete properties, think e.g. wave-particle duality. The same thing is going on with space-time.
Let's consider a very simple thought experiment – imagine a quantum theory of space-time that has a minimal length $l_P$. Moreover, all lengths in the theory can only be integer multiples of $l_P$ (this is just a toy example, the formulas from LQG are similar in spirit but more complicated):
$$ l = n l_p, \; n \in \mathbb{Z}, \; n \ge 0. $$
Naively, this violates Lorentz invariance. For example, if we boost this length by going to a moving reference frame, we expect it to Lorentz-contract according to
$$ l' = \sqrt{1 - v^2} \cdot l, $$
where the value of the square root is continuous hence it can't be consistent with the discreteness of length...
That, however, is completely wrong.
Allow me an analogy to demonstrate the flaw in the argument above. Consider a spinning particle with angular momentum $J = j \hbar$. It is well known that the $z$-component of the angular momentum can only take discrete values ranging from $-j$ to $j$. Does this mean that rotational symmetry is broken? Not at all! There are still continuous rotations around, for example, the $x$-axis acting on the system, these are generated by $e^{i \varphi L_x}$ where $\varphi$ is the rotation angle and $L_x$ is the generator of the symmetry in the spin-$j$ representation. These symmetries act on states by changing the wavefunction components, but they don't change the discrete spectrum. The expectation values of observables therefore transform under continuous rotations continuously, while the spectrum remains discrete.
A similar situation happens with the length spectrum.
Let's denote the quantum states of spacetime with the length under considerations taking values $l = n l_P$ by $\left| n \right>$. We can define the length operator via
$$ L \left| n \right> = n l_p \left| n \right>. $$
Real states are always superpositions of form
$$ \left| \Psi \right> = \sum_n C_n \left| n \right>. $$
Imagine acting with a Lorentz boost on a state of this form. The generator of the boost will continuously change the values of $C_n$, but it won't touch the spectrum.
Alternatively, in the "Heisenberg" picture the state doesn't change at all, but the operator $L$ evolves continuously according to
$$ i \frac{\partial}{\partial \varphi} L = \left[ L, K \right], $$
where $K$ is the boost operator.
In either case, the expectation value contracts continuously:
$$ \left< \Psi' \right| L \left| \Psi' \right> = \left< \Psi \right| L' \left| \Psi \right> = \sqrt{1 - v^2} \cdot \left< \Psi \right| L \left| \Psi \right>, $$
but the spectrum, including the "length gap" $l_P$, remains unchanged and discrete.
Therefore, the existence of minimal length does not go against Lorentz symmetry in the quantum theory of gravity.
At least not in this primitive way. Global Lorentz symmetries indeed don't exist in LQG, but that's not related to discreteness. In fact, global Lorentz symmetries also don't exist in classical General Relativity, unless unphysical constraints of asymptotic flatness are applied.
Now to come to your questions.

What is "between" two atom of space-time or atoms of space (and time) if different? Vacuum? Can distances or neighbourhood be defined if no space and no time and no field is defined?

You'll need to study LQG to answer this question, but I'll try to give you a picture that emerges from applying loop quantization to General Relativity. It may appear superficial, so keep in mind that this structure isn't among the axioms of the theory, instead it can be obtained by a calculation.
The quantum states of spacetime in LQG are very mysterious and still ill-understood. Those can be defined by considering a kernel of the so-called "Hamiltonian constraint operator", defined on another auxiliary Hilbert space called the kinematical Hilbert space (because it doesn't know about the dynamics of General Relativity).
The kinematical Hilbert space $\mathcal{K}$ describes the quantum states of spatial geometry unconstrained by General Relativity. It is well understood and possesses a unique structure.
The basis of states on $\mathcal{K}$ is given by spin networks.
Those are 4-valent graphs (each node has 4 links adjacent to it), where links are labeled by irreducible projective representations of the "little group" $SO(3) \sim SU(2)$, which are just spins, i.e. half-integers $j$. The appearance of the little group has to do with the fact that states are defined at the boundary and not in the bulk, in fact, there's a slight resemblance with the holographic principle here. The nodes of the spin network are labeled by normalized intertwining operators, which are the $SU(2)$-invariant subspaces of $\mathcal{H}_{j_1}\otimes\mathcal{H}_{j_2}\otimes\mathcal{H}_{j_3}\otimes\mathcal{H}_{j_4}$ (here $\mathcal{H}_j$ is the spin-$j$ irrep of SU(2), and $j_{1\dots4}$ are the spins of the 4 links adjacent to the node).
To every surface $S$ immersed in the 3-dimensional boundary, General Relativity associates a geometric area. For example, in classical General Relativity,
$$ \mathcal{A}(S) = \intop_{S} d^2 x \sqrt{g'}, $$
where $g'$ is the induced metric given by
$$ g'_{uv} = \frac{\partial X^{a}}{\partial x^{u}}\frac{\partial X^{b}}{\partial x^{v}} g_{ab}(X(x)). $$
In Loop Quantum Gravity, $\mathcal{A}(S)$ becomes a self-adjoint operator on $\mathcal{K}$. The spin network basis is particularly useful, because spin networks diagonalize area operators. In particular, the eigenvalue of area of a surface $S$ on the spin network state $\left| SN \right>$ is
$$ \mathcal{A}(s) \left| SN \right> = 8 \pi l_P^2 \gamma \sum_{n} \sqrt{j_n (j_n + 1)} \left| SN \right>.$$
Here $l_P$ is the Planck length, $\gamma$ is the LQG-specific Barbero-Immirzi constant which is dimensionless and takes values of order $\gamma \sim 1$, and the sum is over the links of the spin network that intersect $S$.
In LQG, area is quantized. The area spectrum is discrete. The whole spacetime is arranged such that you can't get a value of area which doesn't belong to the spectrum. This is in no contradiction with relativity, for reasons outlined above.
The minimal "area gap" that any physical surface can have is when among the links that intersect it all have spin $0$ (which is equivalent to saying they don't physically exist, because they don't contribute to physical area) except for one which has spin $1/2$:
$$ \Delta \mathcal{A} = 4 \sqrt{3} \pi \gamma l_P^2. $$
If we substitute the value of $\gamma$, fixed by matching the numeric coefficient of the predicted black hole entropy with the Bekenstein's formula:
$$ \gamma = \frac{\ln 2}{\sqrt{3} \pi}, $$
we get a distinctive prediction for the area gap:
$$ \Delta \mathcal{A} = \left( 4 \ln 2 \right) l_P^2 \approx 2.77 l_P^2. $$
The nodes of the spin network can be interpreted as quantum tetrahedra, which are joined along common triangles – the links of the spin network. The areas of triangles are encoded by the spins, and the volumes of the tetrahedra are encoded by the intertwining operators.
In reality (according to LQG), however, space is not a spin network, but a superposition of spin networks. It is easy to see – classical tetrahedra have 6 geometric degrees of freedom (6 lengths), but in LQG there's only 5 (4 spins and 1 intertwiner). Hence, quantum tetrahedra are always fuzzy. Geometry itself is noncommutative. Real tetrahedra on large scales are given by specific superpositions of spin networks that minimize the product of uncertainties between the last remaining 2 degrees of freedom of the tetrahedron (the volume and the dihedral angle). They are called Livine-Speziale coherent states.

Vacuum is generally believed to be the main lowest state of field theory, on space-time. Supposing there is no space-time, can we even define what vacuum is? Does vacuum exist without reference to a particular background independent theory?

Short answer is – no, the vacuum does not exist. The notion of energy does not exist as well (this is already apparent in GR with all of its energy paradoxes – it is possible to define gravitational energy only if GR is expanded around the flat space, which in turn excludes a lot of interesting solutions e.g. the cosmological FLRW solution).
The dynamics of background independent theories is drastically different from anything else. It is in fact completely encoded in terms of constraints – for LQG this is the Hamiltonian constraint.
It is expected (and in fact numerical simulations suggest this is true, see Rovelli's book for references) that among the solutions of the constraint there are those resembling classical geometries satisfying Einstein's equations. Among those, there should be the Minkowski space somewhere.
In fact, there are two formulations of the Hamiltonian constraint operator that are presently known.
One is the canonical formulation, which is defined in terms of matrix elements of the Hamiltonian constraint (or the so-called master constraint) on spin network states. This one is mathematically well-defined, but so far no one was able to prove that it gives General Relativity in the classical limit (and as far as I know there's indications that it may not be true).
The other is the covariant formulation. Here in the spirit of path integrals, the projector on the subspace of solutions of the Hamiltonian constraint is defined in terms of sums over histories of spin networks. These are 2-complexes known as spinfoams. Links of the spin networks trace faces of spinfoams, nodes of the spin network trace edges of spinfoams, structural changes in the topology of the spin networks are encoded in the vertices of spinfoams. The spinfoam model for 4-dimensional LQG is called the EPRL model. In sharp contrast with the canonical formulation, it is not known if this model can be made mathematically well defined (amplitudes for individual spinfoams are always approximate, to get the precise answer we'd need to take the projective limit, for which it is unclear whether it has the right properties or even if it exists). However, it gives classical General Relativity in the classical limit with Livine-Speziale coherent states.
To summarize, LQG is a toy example (which also has the potential to become realistic at some point) of truly quantum space-time. It looks very weird to a physicist who's studying it for the first time. The geometry itself is fuzzy and non-commutative. There is no time evolution, no well-defined notions of conserved energy, no unitarity. This, however, doesn't indicate a flaw in the formulation of the theory (not that there aren't any – there are plenty of flaws in the current understanding of LQG dynamics, but this isn't one of them). Instead, this is an indication that we should use completely new techniques to extract physical predictions. All physics is encoded in constraints, there are no evolution laws. But that also doesn't mean that the theory doesn't incorporate time evolution – it does. Only quantum things evolve with respect to each other, not with respect to an external time flow like in ordinary quantum field theories.
This is very strange and counter intuitive, and we should not have expected any less from a theory of quantum gravity.
