While I am aware of some facts about domain of dependence properties for hyperbolic PDEs, I don't think most of them will be useful for you. The problem is that what you consider as hyperbolic (in your footnote) is too large of a class of equations for the notion to be useful: an illustration is the Heat equation. It is usually classified as a parabolic equation, but it does admit a well-posed initial value problem. So by your definition is hyperbolic. Now it is well known that the heat equation has infinite smoothing properties and infinite speed of propagation.
Now, in the special case of symmetric hyperbolic systems, even in higher orders, one can generally describe the domain of dependence by considering the characteristic cones (see, eg. https://encyclopediaofmath.org/wiki/Characteristic ) of the system. Essentially the "fattest cone" gives you the maximum speed of propagation (which may depend quasilinearly on the solution) and integrating back this cone gets you the domain of dependence.
The domain of dependence properties are really closely associated to a priori energy estimates (see Courant and Hilbert, Methods of Mathematical Physics).
But I don't think this will solve your problem since I don't believe your question can be recast in a form in which such estimates are available.
In particular, looking directly at your equation, on the spatial side it has potentially infinite speed of propagation since the spatial propagation is essentially just a transport equation. So if $W(x,p,t)$ has non-compact support, then the spatial propagation can have arbitrary large speeds. So if your potential vanishes or if your initial data is homogeneous in momentum, your solution will have, as its spatial domain of dependence given by the largest and smallest momentum at which $W$ is supported.
Assuming $f'' = 0$, then you equation can be solved by the method of characteristics: $\partial_t W = v\cdot\nabla W$ where $v(x,p) = (p,f(x))$ is a vector field. The domain of dependence for this problem can be easily found by integrating the vector field independently of the function $W$. I don't know how to deal with your third order term.
Like I said, in general there are only two ways to study domain of dependence properties that are well established, the first is via explicit notion of the Green's function, the second is energy estimates.
PDE books often discuss classification, but they always restrict attention to the case of second order equations, especially for one function of several variables, with good reason. The point of a classification is to find categories of PDE whose analysis has many common features, but there really isn't any general classification in that sense, since the world of PDE is a huge zoo (once you leave the 3 familiar families of elliptic, hyperbolic and parabolic). Think about how you would define parabolic PDEs, even in second order. You already need to look beyond the symbol to distinguish $\partial_t u=\partial_{xx} u$ from $0=\partial_{xx} u$. As the OP points out, the symbol is certainly an important part of the ``classification''. The symbol is only a part of the tableau, which gives a little more information in an algebraic format; see the book of Bryant, et. al, Exterior Differential Systems. But systems of differential equations with the same tableau often have different analysis. Think about the famous Lewy counterexample. There are so many very different genera of animals in the zoo, and broad classifications don't give us much insight. Also look at Gromov, Partial Differential Relations, for lots of examples of PDEs that are locally the same, but globally very different, and are nothing like elliptic, hyperbolic or parabolic.
So question 1: yes, question 2: hyperbolic is tricky to define beyond second order, because already for second order, hyperbolic is very different from ultrahyperbolic, so you really need something to distinguish a Lorentzian geometry from a more general pseudo-Riemannian geometry. On the other hand, your definition of ellipticity is perfect, and does give us some tools to carry out analysis. question 3: a little bit like yes, in that each PDE system gives rise to an algebraic variety, but finally no in that the classification of constant coefficient PDE systems is much finer than the classification of their symbols (it is in fact exactly the classification of their tableau), question 4: yes, you prolong until you hit involution, and so the classification of involutive tableau is not known, a huge messy algebra problem, question 5: like biology, it is messy because there are too many very different animals.
Best Answer
Question 2 is getting clearer now. My sources are Parseval's article from 1800, Poisson's memoire from 1819, Hadamard's Lectures on Cauchy's Problem in Linear Partial Differential Equations (1923), and Baker and Copson's The Mathematical Theory of Huygens' Principle (1939).
On page 133 of the afore-mentioned memoire, Poisson gives the 3-dimensional formula
$$ u(x,t) = t M_{x,t}u_0 + \partial_t (t M_{x,t}u_1); \qquad u_0(x) := u(x,0), \quad u_1(x) := \partial_tu(x,0), $$
where $M_{x,t}g$ is the average of $g$ (defined in $\mathbb{R}^3$) over the sphere centred at $x$ of radius $t$. Then he goes on to prove it, and by the method of descent, derives several special cases, including the 1 and 2 dimensional formulas. So the 3D case is due to Poisson.
Later in 1882, Kirchhoff published a more general formula expressing $u(x,t)$ in terms of the values, the normal and time derivatives of $u$ over an arbitrary closed surface containing $x$, therefore mathematically justifying the Huygens principle. The analogue of Kirchhoff's 1882 formula for 2 dimensions was published by Volterra in 1894. These developments were closely related to the discoveries of fundamental solutions of the Helmholtz equation in 3 dimensions by Helmholtz in 1859, and for 2 dimensions by Weber in 1869.
As for who was the first to discover the 2 dimensional analogue of Poisson's 1819 formula, when he coins the term "method of descent", Hadamard notes
and cites Parseval's afore-mentioned article of 1800, Poisson's memoir of 1819, and Duhem's book from 1891. After giving the 2D formula on page 141 of his memoir, Poisson cites Parseval's article, and says something like "Parseval previously integrated this equation but in a less simple way". Parseval seems to give the formula on page 519 of his article, but I don't understand sufficiently to say the formula is complete. In particular there seem to be no explicit formulas for the quantities Q and Q'. So the 2D case can be said due to Parseval-Poisson.