Let me change your notations. You deal with a 1D quasilinear system with size $N=4$: the standard Cauchy problem is $$ \frac{\partial u}{\partial t}+A(t,x,u)\frac{\partial u}{\partial x}= f(t,x),\quad u(t=0,x)=u_0(x), $$ where $t\in \mathbb R$ (time variable) as well as $x$ (this is a 1D problem), $u$ is valued in $\mathbb R^N$, $A$ is a real-valued $N\times N$ matrix. You may also assume that $A$ depends smoothly (or even analytically) of its arguments.

(1) Your first case: all eigenvalues of $A_0:=A(0,x_0, u_0(x_0))$ are real and distinct, this is indeed the strictly hyperbolic case. In this case, you can guarantee local existence, uniqueness and continuous dependence on the data, i.e. local well-posedness. Of course you cannot expect global existence in general because of the nonlinearity (think about the scalar Burgers).

(2) Let me skip some of your cases and go directly to the case where $A_0$ has a non-real eigenvalue (and thus a pair of non-real eigenvalues). Big trouble ahead: even if the matrix $A$ is analytic, in which case, Cauchy-Kovalevskaya theorem is providing a local (unique) analytic solution, that solution is very unstable in the Hadamard sense. It means that even though $v_0-u_0$ is very small in a very strong topology, such as the $C^\infty$ topology or the $H^s$ topology for a very large $s$, you will not be able to control $u(t)-v(t)$ in a quite weak topology such as $L^2$ (all this is local of course). You will find precise statements in a paper by Métivier, Remarks on the well-posedness of the nonlinear Cauchy problem, with MR number MR2127041.

(3) When all eigenvalues are real, some with multiplicity larger than 2, then instability could or could not occur, depending on other structural factors such as semi-simplicity of $A_0$. Generally speaking, multiple roots will trigger difficulties.

(4) A very important class of theorems, with the name of Lax-Mizohata theorems is establishing a weak converse to (1): if the problem is well-posed (e.g. meaning that you have some Sobolev norm control of $u(t)$ by some Sobolev norm of $u(0)$), then it implies that the system is weakly hyperbolic (case (3)$\cup$(1)). So if you expect your system to be well-behaved in the sense of Hadamard, no choice, the roots must be real-valued, possibly with multiplicity.

Do not think that all physically relevant problem of that type are hyperbolic: to quote just one example, Van der Waals classical system is $$ \partial_t u+\partial_x v=0\quad \partial_t v+\partial_x q(u)=0. $$ When $q'(u)>0$, you are in a hyperbolic region, but when $q'(u)<0$, you have a non-real eigenvalue.