Since I didn't want to think about permutations too much, here is an answer about relating Coxeter transformations of the form $C(A, \text{id})$ to Coxeter transformations of finite-dimensional path-algebras.
Let $A$ be a generalized Cartan matrix. Let $A_+$ and $A_-$ be defined by
$$\left( A_+\right)_{ij} = \begin{cases} A_{ji}, \ i > j \\ 1, \ i = j \ \ \ \ \ \ \ \ ; \\ 0, \text{else} \end{cases} \ \ \ (A_-)_{ij} = \begin{cases} A_{ij}, \ i > j \\ 1, \ i = j \\ 0, \text{else} \end{cases}$$
Theorem 1: We have $C(A, \text{id}) = -A_{+}^{-1}A_{-}^t$.
Proof: This is Theorem $2.9$ in this paper by Sefi Ladkani. Note that I changed the definition of $A_+$ and $A_-$ to match the definition of a Coxeter transformation I gave in my question. $\square$
Now let $H = kQ$ be the path algebra of the finite quiver $Q$ without oriented cycles. Let $S(1), \dots, S(n)$ be the simple modules, ordered in such a way that there is never an arrow in increasing direction, i.e. whenever $1 \leq i \leq j \leq n$ then there is no arrow $i \to j$ in $Q$. This can be achieved by labeling a sink of $Q$ with number $1$, a sink of the remaining quiver after killing $1$ with number $2$ and so on. The homological bilinear form $\left\langle - , -\right\rangle : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$ is given (since $H$ is hereditary) on dimension vectors by \begin{align*} \left\langle \underline{\dim} S(i), \underline{\dim} S(j)\right\rangle & = \dim_k \text{Hom}_H(S(i), S(j)) - \dim_k \text{Ext}_H^1(S(i), S(j)) \\ & = \delta_{ij} - \#\{\alpha: i \to j\}. \end{align*} It is well known that the Coxeter transformation $\Phi_H$ of $H$ is the unique Coxeter transformation of the homological bilinear form, in the sence of Ladkanis paper (i.e. $\left\langle x,y \right\rangle = - \left\langle y, \Phi_{H}x \right\rangle$). Therefore, if we set $D$ the matrix with entries $(D)_{ij} = \left\langle \underline{\dim}S(i), \underline{\dim}S(j)\right\rangle$ then we get $\Phi_H = -D^{-1}D^t$. Furthermore, the matrix $A = D + D^t$ is a symmetric generalized Cartan matrix, and since $D$ has no nonzero entries at the upper triangle (remember the order of the simple modules), we get $A_+ = D = A_-$. We get the following:
Theorem 2: $\Phi_H = C(A, \text{id})$.
Proof: By theorem $1$ we have $C(A, \text{id}) = -A_+^{-1}A_{-}^t = -D^{-1}D^t = \Phi_H$. Compare also with Corollary $2.11$ in Ladkanis paper. $\square$
Remark: I'm pretty sure we can generalize this to every permutation. Also note that we can in this way represent every Coxeter transformation of a symmetric generalized Cartan matrix as the Coxeter transformation of a path algebra (since the numbers in the lower triangle show us exactly how many arrows we have to put between the indices).