The thing about the speed of light $c$ is that it's not just a number associated with a certain type of particle. While we could talk about the mass of the proton, and there would be no problem assuming non-protons had greater or lesser masses, the value $c$ is an entirely different beast.
$c$ is an intrinsic property of spacetime itself, not of the particles in spacetime. You wouldn't expect there to exist anything (particle, signal, information) that you could insert into spacetime with the property "changes the nature of spacetime for itself."
The above is meant to undermine the "if it's just a property of everything we've found so far, we haven't ruled out finding things without this property" line of thought.
For a more concrete demonstration of what goes wrong when you alter the structure of spacetime as we know it, take a look at the "tachyonic antitelephone". This demonstrates how any abstract communication faster than light leads to causality violations as observed by even not-faster-than-light observers.
There are many variations on the thought experiment; here's one of them (with the math worked out in the linked article): $A$ is moving away from $B$ with speed $v < c$ in $B$'s reference frame. $A$ sends a faster-than-light message to $B$, who responds in kind with a reply. The problem is for sufficiently fast (but not faster than $c$!) speeds $v$, $A$ will receive the reply before sending the message. Faster-than-light anything begets time travel.
Since it gets repeated far, far too often, I'll also counter quantum entanglement arguments here. Quantum entanglement does nothing in the hypothetical "why can't this communicate faster than light?" scenarios other than guarantee that the two particles will "collapse" to the same (or opposite, or orthogonal, or whatever) states when either is observed. The only quantum mechanical aspect of the whole thing is the fact that the eventual collapsed state can't even exist before the measurement (it's not a hidden variable).
But the correlation -- the thing you want to rely on to communicate faster than light -- could be achieved entirely classically. Take a red marble and a blue marble, put them in a bag, and draw one out randomly without looking. Lock your marble in a box, and hand the bag to someone else, who also doesn't peak. Send the person far away. Then, look at your marble. If it's blue, you instantly know the other person has red. But you transmitted nothing. All quantum mechanics does is make it so who has which marble isn't pre-determined.
This shouldn't come as a surprise, because the only notions of space and time in quantum mechanics are the ones from spacetime itself. Quantum mechanics doesn't come equipped with some independent notion of distances. So whatever structure spacetime has applies to quantum things just as much as marbles and people and signals.
Everything said here stays true even in general relativity, by the way. Enabling things to move faster than $c$ still leads to causality violations. Even wrapping things in a "warp bubble" and moving that faster than $c$ leads to causality violations.
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.
Best Answer
Yes.
Yes, but note that $c$ is a universal constant. If something is traveling at the speed of light, it is traveling at the speed of light to everyone (except other photons traveling parallel, see my answer to this question).
It boils to relativity. The energy of a particle is related to the velocity, $v$, via the relation $$ E=\gamma mc^2=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} $$ As $v\to c$, $E\to\infty$, but gives an indeterminate operation at $v=c$.
(a) 50 mph is a speed, not an acceleration and (b) you can't get somewhere "50 miles early." If your trip is 300 miles, you can't get there in 250 miles. You can get there in a shorter amount of time.
The 2nd postulate of special relativity states that $c$ is invariant of reference frame (constant for everyone), so nothing can accelerate to speeds faster than light.
Faster than light means that you have a speed $v>c=2.9979\times10^{10}$ cm/s. If you have done this, then kinematics suggest you have a complex velocity (as in the imaginary number complex) which is utter nonsense since velocity is a physical (real) quantity.