What does this notation $g(x) = S\{\delta(t) \}(x)$ mean?
My understanding is that a system $S$ is a notation for   a generic object, which takes instances depending on its arguments, given between $\{\cdot\}$. Here, $S\{\delta(t)\}$ instantiates it as its effect on the zero-centered impulse response, with free variable $t$, taken on variable $x$. 

Now, if this system $S$ is linear and shift-invariant, the operator given by $S\{\delta(t)\}$ defines $S$ it completely. 

If the system applies on discretized inputs (that we can call $x$), it can be rewritten as a linear operator, represented by an infinite Toeplitz matrix. Each row implements the (linear) convolution, and the Toeplitz structure emulates the shift-invariance. I suggest to read Toeplitz and circulant matrices by R. Gray. The $0$ in $A_{x,0}$ may refer to the way an infinite matrix is matched with a signal centered at $0$, but I still wonder if this notation is clear.