It sounds like your confusion is coming from taking paraphrasing such as "everything is relative" too literally. Furthermore, this isn't really accurate. So let me try presenting this a different way:
Nature doesn't care how we label points in space-time. Coordinates do not automatically have some real "physical" meaning. Let's instead focus on what doesn't depend on coordinate systems: these are geometric facts or invariants. For instance, our space-time is 4 dimensional. There are also things we can calculate, like the invariant length of a path in space-time, or angles between vectors. It turns out our spacetime has a Lorentzian signature: roughly meaning that one of the dimensions acts differently than the others when calculating the geometric distance. So there is not complete freedom to make "everything" relative. Some relations are a property of the geometry itself, and are independent of coordinate systems. I can't find the quote now, but I remember seeing once a quote where Einstein wished in reflection that instead of relativity it was the "theory of invariants" because those are what matter.
Now, it turns out that the Lorentzian signature imposes a structure on spacetime. In nice Cartesian inertial coordinates with natural units, the geometric length of a straight path between two points is:
$ds^2 = - dt^2 + dx^2 + dy^2 + dz^2$
Unlike space with a Euclidean signature, this separates pairs of points into three different groups:
$> 0$, space like separated
$< 0$, time like separated
$= 0$, "null" separation, or "light like"
No matter what coordinate system you choose, you cannot change these. They are not "relative". They are fixed by the geometry of spacetime. This separation (light cones if viewed as a comparison against a single reference point), is the causal structure of space time. It's what allows us to talk about event A causing B causing C, independently of a coordinate system.
Now, back to your original question, let me note that speed itself is a coordinate system dependent concept. If you had a bunch of identical rulers and clocks, you could even make a giant grid of rulers and put clocks at every intersection, to try to build up a "physical" version of a coordinate system with spatial differences being directly read off of rulers, and time differences being read from clocks. Even in this idealized situation we cannot yet measure the speed of light. Why? Because we still need to specify one more piece: how remote clocks are synchronized. It turns out the Einstein convention is to synchronize them using the speed of light as a constant. So in this sense, it is a choice ... a choice of coordinate system. There are many coordinate systems in which the speed of light is not constant, or even depends on the direction.
So, is that it? It's a definition?
That is not a very satisfying answer, and not a complete one. What makes relativity work is the amazing fact that this choice is even possible.
The modern statement of special relativity is usually something like: the laws of physics have Poincare symmetry (Lorentz symmetry + translations + rotations).
It is because of the symmetry of spacetime that we can make an infinite number of inertial coordinate systems that all agree on the speed of light. It is the structure of spacetime, its symmetry, that makes special relativity. Einstein discovered this the other way around, postulating that such a set of inertial frames were possible, and derived Lorentz transformations from them to deduce the symmetry of space-time.
So in conclusion:
"If all motion is relative, how does light have a finite speed?"
Not everything is relative in SR, and speed being a coordinate system dependent quantity can have any value you want with appropriate choice of coordinate system. If we design our coordinate system to describe space isotropically and homogenously and describe time uniformly to get our nice inertial reference frames, the causal structure of spacetime requires the speed of light to be isotropic and finite and the same constant in all of the inertial coordinate systems.
Relative velocity for sound waves is not a "symmetric" situation. For example, in the extreme case of a fighter jet approaching a stationary observer at Mach 1, the jet will be traveling as fast as its sound waves. The observer will not hear anything until the jet gets to his position.
On the other hand, for the case of a stationary sound source and an observer moving toward the sound source at Mach 1, the observer will obviously hear a doppler shifted sound of a much higher frequency than what the sound source is emitting.
Such a non-symmetric situation requires the last formula listed in the question, where corrections are made for both the velocity of the source and the velocity of the observer.
Best Answer
A modern view in physics is that when we define a term such as "stationary," we should attempt to do so operationally meaning in terms of some sort of a measurement one can perform. Before we start thinking about the existence of a stationary frame in the real world, we need to define such a frame operationally. Once we have a definition that is sufficiently operationally precise, we can go out into the world and make measurements to see if there exists anything out there that satisfies our definition.
I would recommend that you try to define "stationary frame" in some operational way such that the resulting definition aligns with your intuition for what stationary should mean, and I'll bet you won't be able to do it.
Fortunately, there is another term in physics that is pretty close to what you might want, the notion of an "inertial frame." An inertial frame is one in which if you were holding an accelerometer still in that frame, then the accelerometer would indicate zero. So basically, such a frame is one which is not accelerating. What's interesting about inertial frames, however, is that if you are in an inertial frame, then any other observer moving relative to you with constant velocity will also be in an inertial frame. In other words, if that other observer were holding an accelerometer still relative to him/herself, then he/she too would measure zero acceleration. In such a situation, you might be inclined to claim that you are the one standing still, and the other observer is moving. However, it is a basic fact of physics that the laws of physics are the same in all inertial frames, so from an operational viewpoint, there is no way to distinguish between inertial frames.
It is in this sense that saying that one inertial observer is stationary while another is moving is not really appropriate. It is also because of this that you will often hear statements like "there is no absolute rest" or "motion is relative" or "it only makes sense to talk about relative motion" etc.
The bottom line is that all frames can be categorized as either inertial or non-inertial. All inertial frames are physically equivalent and are the closest you can get to being "stationary" in some sense, while non-inertial frames are pretty far from what most people would call stationary since acceleration is involved.
I hope this helps! Let me know if you want more! (I know I didn't address the questions about distortions etc., but I felt the answer was getting too long).
Cheers!
Addition. Just for the sake of more completeness; it's important to note that infinitely large frames of reference that are inertial don't actually exist. There are, however, "local inertial frames," namely it is possible to devise a physical situation in which a small region around you satisfies the definition of an inertial frame outlined above, but as you move further away from this region, the frame becomes a worse an worse approximation to being inertial. I'd encourage you to explore this further since this point is especially important in GR for example. Here and here would be good places to begin to learn more.