Note that for cosmic expansion, special relativity only applies when the spacetime region in question is approximately flat, which in our case happens when it is small.
As the M31 galaxy is moving toward us at great speed it's "depth" should appear slightly flattened for us. A sphere moving toward us at the speed of light will appear "pancaked" in lack of a better word.
No, that's not correct. You have a worldline in spacetime, and its direction at an event is your four-velocity. The orthogonal complement of your four-velocity, i.e., all directions perpendicular to it, is your space. What Lorentz-FitzGerald length contraction says is that if you project a moving sphere onto your space, the result is contracted along the direction of motion.
The difference between this and what you say is that there is no mention of how the sphere will appear. The sphere is contracted in the inertial frame comoving with you at that instant. It doesn't appear contracted--in fact, if you took a picture of it with a hypothetical super-fast camera, the relativistically moving sphere will appear... still completely spherical. But not quite the same as a non-moving sphere if it has features on the surface, as Penrose-Terrell rotation makes some of its "back parts" visible.
Does this also mean that electromagnetic radiation (light) from a flattened star is focused in the direction of motion relative to the observer?
Yes. Just as a vertically falling rain comes at an angle when you're riding in a car, the angles at which any signal comes are different in a moving frame compared to a stationary frame. This is described by relativistic aberration. Additionally, Doppler shift changes the intensities of a radiating source along different directions, making the overall effect relativistic beaming.