Is space infinitely divisible? As a child I remember hearing the popular paradox presented by Zeno proposing that Achilles could never catch a tortoise in a race since he would have to traverse the infinite space between himself and the tortoise.  After some pondering I concluded that the obvious answer to the dilemma is that the space is not infinitely divisible.  Although in later years I learned that this paradox can be explained by a finite infinite series, I have always found it difficult to accept that just because a concept can be supported mathematically, it must therefor exist in reality.  In the case of Zeno’s paradox, although finite infinite series do exist, I believe the simple explanation I constructed as a child to be a better answer since it is both a viable solution to the problem, while also being comprehensible at a practical and intuitive level that is often lost with mathematical constructs.  A few years ago I learned that the current scientific position is that space is in fact infinitely divisible and I have been disturbed ever since.  
Does Infinity exist?  In the literature the overwhelming answer to this question by the scientific community is yes... on paper.  It is interesting to consider that in spite of a prolonged search for infinity in nature, it remains elusive.  Although the importance of the concept of infinity in the field of mathematics is indisputable, ironically, it represents perhaps the most compelling example of a mathematical constructs that is “true” yet non existent in reality.  Although many will argue that infinity exists in the singularity at the heart of a black hole, or perhaps at in some fields, when these phenomena are examined they become logically incomprehensible, mathematically error ridden, and physically non existent.  If one were to weigh the evidence with respect to the physical existence of infinity, the overwhelming weight points to its non existence.  If infinity doesn’t exist than space must be finitely divisible.  If so, of what is space divisible into, and what are the consequences of its finite divisibility?
In searching for a possible constituent of the fabric of space one could consider whether the oscillations of electro magnetic states observed by the motion of matter/energy through space are a reflection of the matter/energy “train” moving through space, or perhaps an illumination of the “tracks” on which it travels.  As a photon of light, buckyball, or large celestial body travels through space, could it be passing through (occupying), a mesh like network of electro magnetic “cells”?  These cells could consist of an electro core surrounded by a magnetic space that causes oscillations in state as the matter moves from one cell to another.  Perhaps this process is similar to the duel particle process by which electrons jump between orbitals... i.e.) The energy is particulate in nature while in the middle of each cell but it never occupies the space between each cell as it moves from cell to cell... instead it is exhibits wave like properties between cells.   
Dr Richard Feynman once stated that the double slit experiment contains the only mystery of quantum mechanics.  If one considers the double slit experiment from within the framework of an electro magnetic mesh through which matter and energy travel the experiment may be able to be accounted for in a more conventional manner than current postulations.  Imagine the borders of the electro magnetic cells constantly moving in relation to the material containing the double slits as the earth, laboratory, apparatus, and particles being fired at the screen constantly move through the mesh like network.  Depending on the position of the cell abutting the screen relative to the slot, the particle could be in a state of solid, defined, particulate matter, or in a state of diffuse energy (electro or magnetic).  If the particle hits the slot in the screen in a state of diffuse energy it will diffusely enter both slits and interfere with itself.  If it hits the screen during the solid particle phase it will pass through one of the two slits or hit the barrier and travel through none at all.
Although I am not able to support these ideas mathematically due to a lack of training, the model I am proposing has some interesting intuitive possibilities.  These include; 
In such a model could movement from cell to cell produce the electromagnetic radiation seen as matter changes velocity/direction when acted on by a force, or by the oscillation of atoms?  Perhaps there is an electro magnetic “lag” of the vector a particle was on before force was applied to change its direction.   
Could the motion of matter through a network of electromagnetic cells produce gravitational forces?  Interesting... the more matter massed together the more cells being “activated” the more gravity being produced.  If this was true one would also expect an increase in velocity to result in an increase in gravitational forces since more cells would be activated per unit time.  
Could time itself be a function of motion through electro magnetic cell with time being relative to motion through cells similar to the example of gravity mentioned above? 
Perhaps some of these questions have been addressed with research done on electron orbitals and energy states?
 A: The question kind of jumps around a bit, but I'll try to answer what I can in an intuitive and not complicatedly mathematical way.
Starting with the main issue at hand, is space infinitely divisible. I am going to say yes, absolutely. To prove it, allow me to present a relatively simple logical argument. Let us suppose space were not infinitely divisible. That would mean that in a given inertial reference frame, there must be a smallest possible distance. Furthermore, it would mean that all motions within said reference frame must be integer multiples of this smallest distance, otherwise I could travel a non-integer multiple forward and an integer multiple backward and wind up at a distance less than the smallest one from my starting point. Because space is 3-dimensional, I can naturally assume that moving at right angles to one's current trajectory would be allowed, since it merely represents moving in a different Cartesian direction. So let us picture a fundamental massive particle (I don't care which one)  moving very slowly in this frame. Since any displacements over time or space are limited to integer multiples of our smallest distance, the smallest amount this particle can move is one "square" ahead; let's picture it doing that now. Then, due to thermal motions, our particle moves one square to the right. Now, how would I describe the new location of the particle relative to the old location? One block forward and one block right? I can only say it must be a non-integer multiple of our smallest distance; something between 1 and 2. But hold on, you say, perhaps that's allowable as long as no single motions are non-integer multiples. To that I say this, I never did specify what direction "forward" or "right" relates to. The truth is, there is no privileged directions, so at any given time, that particle should have been able to travel in any direction. That means this particle, to get back to its initial position (a necessarily allowable motion), could move in a straight line for that non-integer multiple distance. Even if we restrict motion to integer multiples, it would travel towards the starting point one block and the net displacement would be less than the smallest distance possible. This means that there must, therefore, be a smaller distance than what we defined as the smallest. But, since we defined no number, there must always be a smaller distance than whatever we choose to be the smallest. Thus, space must be infinitely divisible. Perhaps not infinitely measurable, but definitely infinitely divisible.
I would attempt answering the rest, but that's more of a statement of personal theories. All well and good, but not really addressable in the operational context of this website. Besides, I think my last point renders any answer I give as moot.
A: One cannot do physics without mathematical abstractions, and in particular without the notion of infinity. But infinity cannot be observed - it must be extrapolated from what we observe. Thus all numbers measured are in fact rational, though we know that the diagonal of a square of rational side will be irrational, and hence have an infinite number of digits that is not periodic.
The physical theories that summarize our current theoretical understading of physics (namely quantum field theory and general relativity) assume that space is parameterized by three continuous variables, and hence is in prnciple infinitely divisible. But due to the Heisenberg uncertainty relation, arbitrarily small features cannot be observed except by expending arbitrarily much energy. (This is why the colliders used to probe the detials of the structure of enlementary particles must work with very high energies, and rising costs limit even more what can could be done in principle.)
However there are widespread speculations that in quantum gravity the small-scale structure of space and time should have to change. But due to lack of experiments and the difficuties of theoretical work, no consensus can be expected in the near future, and for a long time to come.
Regarding your ideas: Already to predict the effect of motion through electromagnetic cells requires that you adhere to the standard way of describing electromagnetism, which requires that you accept that space is continuously parameterized. Physics without continuity in space and time is bound to be severely crippled....
A: We have developed an atomist model of the universe with the quantum being the smallest packet of energy. If matter and energy are quantized, one might make the qualitative leap that space is quantized as well. Quantized space does not violate any physical principles. 
To assume that space is infinitely divisible when energy and matter are not seems to be a luxury.
