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 should probably only be defined for 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.
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 should probably only be defined for 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.