Is the matrix $\sum_{g \in G} a_g \rho(g)$ normal and what further properties does it have?

Question

Let $\rho$ be the regular representation of $G$. $S \subset G$ a generating set, $|g| := |g|_S=$ word length with respect to $S$. Then I construct such a matrix, where we have some ordering $g_i$ of the group $G$:

$$ a_{i,j} = \frac{1}{1+|g_i g_j^{-1}|}$$

This is a group matrix as defined by Dedekind and Frobenius. Let $H_G:= \sum_{g \in G} \frac{1}{|g|+1}$ be the harmonic number associated to $S$ and $G$.

Here are my conjectures concerning this matrix some of which I can prove:

1. $H_G = |A|$, where $|A|$ denotes the spectral norm of $A$ [proved with Perron Frobenus theorem]

2. (If 1. is true, then by definition of $A$ we must have that $1/H_G A = 1/|A| A $ is a doubly stochastic matrix [that $\frac{1}{H_G}A_G$ is doubly stochastic and positive is clear from definition, so this is also proved]

3. $A = \sum_{g \in G} \frac{1}{1+|g|} \rho(g)$ is the Birkhoff-Neunmann decomposition induced by the doubly stochastic matrix [ proved, by comment of darijgrinberg]

4. Using 3. I can prove that $A$ is a normal matrix

5. $A$ is non-singular. [that is unclear to me, but numerics suggest that's true]

If we regard 3. as the definition of $A$ then I am able to show that $A$ is normal:

It basically boils down to $\rho(g)^T = \rho(g)^{-1}$ as $\rho$ is a permutation matrix and hence orthogonal and that:

$$\sum_{g,h \in G} \rho(g^{-1} h) = \sum_{g,h \in G} \rho(g h^{-1})$$

However I am a bit unsure about the last step.

For the 1. part: I think of using the Perron Frobenius theorem on the positive and doubly stochastic matrix

$$\frac{A}{H_G}$$

I can prove that

$$A \cdot 1 = H_G \cdot 1$$

where $1$ is the vector consisting of all $1$-s.

For the 5. part:

By a therorem (page 12) of Frobenius, we have:

$$\det(X_{gh^{-1}}) = \prod_{ \rho \text{ irred }}\det(\sum_{g \in G} X_g \rho(g))^{\deg(\rho)}$$

Hence for 5. it remains to show that if $\psi$ is an irreducible representation then:

$$0 \neq \det(\sum_{g \in G} \frac{1}{1+|g|} \psi(g))$$

Maybe this could be proved with Fourier Analysis of finite groups?

Any help for proving 1-5 is highly appreciated. Thanks for your help!

Related: https://mathoverflow.net/questions/330359/what-properties-characterize-the-function-lx-x-expx-logx