What "propagates" a force through the rest of a solid? So, in typing the title of this question I was recommended this awesome one, which confirmed my guess that this effect "propagates" at the speed of sound (though I just had a feeling, I don't really understand why).  I also only have high-school level physics experience.
I feel like I don't know physics well enough to really know what to ask correctly, but I'll try to explain.
So, I rotate my arm, and, even though tendons are just pulling it at a certain location, the rest of it follows suit.  Why is that?  What is happening at an atomic/molecular scale that ends up "conveying forces" over a distance?
And, something else I don't understand- what is so special about the speed of sound that makes it this "fundamental unit of translation", or something?  Maybe the better question is "what processes at an atomic/molecular scale lead to the speed of sound being associated with all of these behaviors", which ties both questions together.
 A: Solids are just a collection of atoms that are bound together chemically, i.e. through the electromagnetic force. They aren't perfectly rigid. Think about a linear chain of atoms. Pushing on the end atom will cause it to get closer to its neighbour. They will repel each other, so the neighbour will then move away, getting too close to its neighbour on the other side. This process continues all the way along the chain of atoms. By the time the last atom has moved, you have moved the entire chain. Pulling on the end of course just has the opposite effect as pushing on it. The same effect is what happens in solids, but instead of a linear chain you have a 3D lattice. Wikipedia has a nice demonstration of this in 2D.

You see that it isn't an instantaneous process. You correctly identify the speed of the propagation of this effect as the speed of sound, $c$. However, this is just how the speed of sound is defined: the propagation speed of a small deformation in the material. This depends on the stiffness (measured by the bulk modulus $K$) and the density $\rho$ of the material:
$$
c = \sqrt{\frac{K}{\rho}}
$$
This can vary a lot. The speed of sound in air at SATP is about 343 m/s, whereas in steel it is closer to 6000 m/s.
