How do atoms in a solid "communicate" force to each other? What is the mechanism that carries and communicates force in a solid, on the atomic level? 
Is there some other mechanism besides atomic deformation and proximity?
That is, if I had an infinitely incompressible substance and put it on top of my hand and hit it with a hammer, would my hand feel anything, if there is no difference whatever in the movement, shape or location of the substance?
(If you think the substance will move down in response to the hammer, remember that it's incompressible, so the top of the substance can't move down faster than the bottom, and the whole thing can't immediately move down, or we would have sent information to the bottom instantly, i.e. faster than the speed of light. How, then, would the information be communicated throughout the substance that it's time to gain downward momentum?)
If you argue that "infinitely incompressible" is a ridiculous scenario to assume, :) because it violates the compressibility of all matter, then is it that basic compressibility of all matter that provides the mechanism by which larger-scale-than-nuclear force is transmitted from atom to atom? In other words, is all force on the larger-than-atomic scale the result of inter-atomic and infra-atomic compression/deformation/shifts in density?
 A: "Infinitely incompressible" implies that the speed of sound / pressure waves is infinite (larger than the speed of light).
But even if you accept that the medium has this property, then the hammer you hit it with might not. The pressure at the point of impact of the hammer corresponds to the fraction of the hammer that is instantaneously seeking the impact (the back of the hammer doesn't initially "know" that the front has hit your incompressible surface.
But getting back to the essence of your question - all material deforms under stress - longitudinally and laterally. The fact that a solid has an initial volume / shape comes about from an equilibrium between attraction and repulsion - sitting at equilibrium (the bottom of a potential well) there will always be a small range over which displacement gives rise to a linear force. 
None of which exactly "answers" your question because it is based on two contradictory premises - I hope it is nonetheless helpful.
A: I will address this part, which is essential in understanding anything further about matter.

What is the mechanism that carries and communicates force in a solid, on the atomic level? 

At the level of collection of atoms all the "forces" are electromagnetic. Atoms are neutral, but they have shapes from the orbitals of the electrons which "orbit" the nucleus, as an example:


The five d orbitals in ψ(x, y, z)2 form, with a combination diagram showing how they fit together to fill space around an atomic nucleus.

These shapes allow the positive charges of the nucleus to spill out and generate attractive forces,  the atoms fitting into  molecular and lattice shapes, conceptually  fitting like Lego blocks, the electric fields defining the lattices and the strength of the solid. 
Any pressure on the lattice will be transmitted by electromagnetic forces, and cannot be  faster than the velocity of light. In fact it is the collective behavior of the ensemble of atoms/molecules that will define the thermodynamic quantities like pressure etc for the solid, an emergent behavior from the collective behavior of the underlying atomic level.
