Is spacetime discrete or continuous? Is the spacetime continuous or discrete?
Or better, is the 4-dimensional spacetime of general-relativity discrete or continuous? What if we consider additional dimensions like string theory hypothesizes? Are those compact additional dimensions discrete or continuous? 
Are there experimental evidences of continuity/discreteness?
When particles move inside space do they occupy spacetime by little chunks?
What would imply if spacetime is discrete on continuous theories?
I've found little information on the web and books.
Probably my question is ill-posed and I apologize for this.
 A: There is a beautiful theory of quantum gravity called "Canonical Quantum Gravity" which aims to quantized general relativity using typical canonical methods (canonical quantization/path integral formulation).  This theory predicts a granular structure to space-time while maintaining local Lorentz invariance.  The theory gives a spectrum of eigenvalues for quantized area and volume based on Penrose's spin-network graphs, except the theory considers equivalence classes of spin-networks under diffeomorphisms.  The Path-integral formulation of the theory consists in considering a sum-over-geometries which is entirely background independent, carried out in sum over 2-complexes, which are themselves graphs.  Here is a small set of lectures that might interest you:
http://arxiv.org/abs/1102.3660
Response to comment by OP: There are no experimental tests of quantum gravity that we know as of now, whether because we don't know how to interpret what we already have in front of us, or because we simply don't have the technical power/creativity yet, although there are a number of new papers that suggest experiments that may be done at the LHC for Canonical Quantum Gravity, which have to do with the evaporation of micro-black holes and their radiation spectra which differs from the classical spectra predicted by QFT in curved space-time.  Canonical Quantum Gravity is also the only mainstream theory of QG on the table that gives falsifiable, numerical predictions that are novel; at least I have yet to see anything else on the forums and arxiv that does, so that doesn't mean much.
A: 
is the 4-dimensional spacetime of general-relativity discrete or continuous?

In the usual definition of general relativity, spacetime is continuous. However, general relativity is a classical theory and does not take quantum effects into account. Such effects are expected to show up at very short distances, where your question is relevant.

Are there experimental evidences of continuity/discreteness?

All the experimental evidence points to continuous space, down to the shortest distances at which we have been able to measure. We don't know what happens at shorter distances. We also do not have any direct experimental evidence that gravity is a quantum theory, with the same caveat.
On the other hand, we are quite confident that a complete theory of nature must include quantum gravity and not just classical gravity. And, we have an educated guess of the distance scale at which quantum effects should become measurable: this is the Planck length, roughly $10^{-33}$ cm. This is much much shorter than the shortest distance at which we can carry out experiments, so at least we are not surprised that we did not see any such effects so far.
Before proceeding, one more caveat. There is an interesting and quite recent astrophysical experiment that showed that Lorentz symmetry holds even below the Planck length. If Lorentz symmetry is broken, it generally means that photons with different energies will travel at different velocities. At the experiment, they managed to detect a pair of photons that were created at almost the same time but had very different energies. They reached the detector almost simultaneously, which means their velocities were similar. Because the photons travelled an enormous distance before reaching us, they must have had almost the same velocity.
So we know that at least Lorentz symmetry holds at very short distances, and it seems difficult to reconcile this experimental fact with a discrete spacetime. So at least naively it seems that this is evidence against discreteness.

Is the spacetime continuous or discrete? 

At long distances spacetime can certainly be thought of as continuous. At short distances, the short answer is: we don't know.
String theory is the only consistent theory of quantum gravity we know of, where we can actually compute things with some confidence. (You will probably hear some opinions that contradict this statement, mentioning loop quantum gravity, causal sets, etc., which are not related to string theory, but what I said is the common view in the community of high-energy theorists.) String theory is giving us some strong hints that perhaps spacetime at short distances is not continuous or discrete, but something else that we don't understand yet.
So the situation is that even theoretically, without talking about actual experiments that check the theory, we don't know what spacetime is like at short distances. Perhaps this is why you don't see this question mentioned a lot. My personal guess is that spacetime at short distances is neither continuous nor discrete, but has a different nature that may require new mathematical tools to describe.

Or better,  What if we consider additional dimensions like string theory hypothesizes? Are those compact additional dimensions discrete or continuous?

Adding extra dimensions does not change any of the above.
A: the idea of spacetime having a fundamental length does not necessarily translate in a discretized structure. 
Let's think intuitively this in term of path integrals (lets assume one-dimensional paths and forget about stringy structure for now, is not relevant to the discussion). When we do path integrals, we usually take all kinematical paths of the system in configuration space (what is usually called off-shell states), assign an amplitude given by the dynamical action, and sum them all to obtain physical observable amplitudes (the on-shell states)
Now, the planck scale sets a natural cutoff for on-shell states, because paths that have energies above that scale must result in black holes in the path (or the quantum gravity equivalent of black holes, whatever those turn out to be). So in your amplitudes for on-shell states, you get systems that do not have observable structure beyond the planck scale, and in fact, increasing the energy makes it worse because it makes the resulting black holes bigger. But they live nonetheless in a Lorentz-invariant background
Now, all this is speculative, and likely not entirely correct picture, but my point is that a finite minimum physical scale does not contradict a continuous Lorentz-invariant background
A: For the four dimensions space-time that we're used to, atoms of space-time is incompatible with special relativity. If we were to try claim a size of these grains of space time we would also have to say in what reference frame in which they have that size. So they introduce a preferred reference frame. 
From my understanding, supersymmetry introduces completely discrete dimensions of space-time, but these are radically different from the dimensions we are accustomed to. Here's a much more better discussion of the topic by one of the leading theorists in the world. 
A: There is an argument known as Weyl's tile argument which is not physics but philosophy, involves some really easy math and accessible to laymen like myself. Still, I'm tempted to put this here since it answers your question even though this probably doesn't belong on a physics forum.
In a discrete space, say a square/rectangular tiled space, (for convenience) we start by constructing two sides of a triangle, each of 1 unit length . To traverse the hypotenuse from either point, we have to move one unit of length to the right (or left) and one unit of length down (or up).

Say AC is traversed in 2 steps, A-D, D-C we have a length of 2 units along AC in the tiled space.
Suppose we keep increasing the number of steps taken from A to C and decreasing the size of unit length, path along AC would look like this :
Length along the zig zag path above AC is still larger than the length of hypotenuse by factor of √2, which was the same factor when we used a much larger unit of space and only 2 steps (n=2) to traverse along the hypotenuse!
This is essentially the Weyl's tile argument 

the former result does not converge to the latter for arbitrary values of n, one can examine the percent difference between the two results: (n√2 - n)⁄n√2 = 1-1⁄√2. Since n cancels out, the two results never converge, even in the limit of large n. 

This tells us that no matter how small a unit of length we take, not even an infinitesimal length, would even approximate the pythagorean theorem in a discrete space.
 It happens to be true because of the simple observation that you have to be able to travel across in space in any direction, which is, in this example, 1/2 to the right & 1/2 to the down (45‎°) simultaneously for a unit , and not a unit towards right then a unit to the down, which is what happens if we discretise length. For the pythagorean theorem to work, a fixed length measured along one direction must not vary when measured along another direction. This is known as isotropy of space, which is a property of the continuum.
Discrete models with different structures other than rectangular can also be disproved using the same argument.
In a sense, this argument doesn't fall prey to unfalsifiable claims that there is discreteness, but beyond our abilities to experimentally observe. It doesn't matter how small the "grains" or "pixels" may be.
Take 3 sticks, two of them having a length of 1 metre and one of approx 1.414 metre, all of them measured along a common axis. Try to make a right triangle, if the hypotenuse falls well short of completing the triangle or after some rotation, extends beyond it, (heh) you're in a universe with discrete space.
About Time
Relativity itself only actually observes that there is “movement”, and “assumes” there is “time”. 
If, I say, for instance “The bus arrives here at 9 o'clock,” I implicitly mean that the pointing of the small hand of my watch to 9 and the arrival of the bus are simultaneous events
This seems perfectly acceptable, unless you realise that we are comparing the co-ordinates( location) of one thing to a thing called “time”.
But in fact, the co-ordinates of one thing (a bus) are only compared to the coordinates of another thing ( the location of a rotating pointer, or the pulse in the circuit, in case of a digital clock ).
The point being, coordinates of space are used to measure time, so one could say they are really the same thing. If space is continuous, so is time. 
A: To answer your question, the spacetime may be continuous or discrete; you can't tell if the mathematics of the latter converges to that of the former. Now, in reference to Weyl's tile paradox I would like to point out the following. What the argument shows is that the distance in the discrete geometry of the grid doesn't converge to the distance in the continuous geometry of the plane under the limit over a sequence of refinements of the grids into smaller and smaller squares. However, the mismatch is caused by choosing -- and keeping -- particular directions for the axes of the grid. It shouldn't be surprising that this produces an anisotropic effect. What if the limit is over an array of grids that not only refines them into smaller and smaller squares but also rotates them through smaller and smaller angles? Then the difference between the distances along the grids and the Euclidean distance converges to zero in the sense of $\lim\inf$. 

I wrote some details here: http://inperc.com/wiki/index.php?title=Convergence_of_the_discrete_to_the_continuous .
A: Since 2014, according to a specific mathematical physics equation and theorem described in: 

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.091302,
  preprint https://arxiv.org/abs/1409.2471 

with proof in

https://link.springer.com/article/10.1007%2FJHEP12%282014%29098,
  preprint https://arxiv.org/abs/1411.0977),

one can argue there exists a geometrical description of Lorentzian spacetimes with 4D, 3D or 2D volumes quantized in integer values of Planckian units.   
Among the physical consequences, these different aspects of spacetime discreteness, provide respectively "quantization of the cosmological constant, mimetic dark matter and area quantization of black holes" according to the authors of the quoted papers: Ali Chamseddine, Alain Connes and Viatcheslav Mukhanov (respectively theoretical physicist, mathematician and cosmologist).  
The detail of the computations regarding the connection between mimetic dark matter and dark energy with discreteness of 3D or 4D volume can be found in https://arxiv.org/abs/1702.08180 
If null results persist in the search for dark matter particles and mimetic gravity phenomenology remains compatible with multimessenger astronomical observations (https://arxiv.org/abs/1811.06830), the discretness of spacetime might emerge as a relevant hypothesis.
One may notice that the high-energy physicist John Iliopoulos who made in 1974 a memorable "Plenary report on progress in Gauge Theories" paving the way to the completion of the current Standard Model of particles (http://inspirehep.net/record/3000/files/c74-07-01-p089.pdf …) has recently reported  that this geometric framework "may offer a new insight for the mysteries of dark matter and dark energy".(https://www.epj-conferences.org/articles/epjconf/abs/2018/17/epjconf_icnfp2018_02055/epjconf_icnfp2018_02055.html) 
Of course this last remark should not be taken as an authoritative argument but aims at showing that this geometric paradigm that is almost orthogonal to the current one in the (astro)particle physics community does not make it an irrelevant one! 
