One has big problems trying to make category $B$ work.

Standard Lefschetz principle arguments justify Lars' use of $\mathbb C$ even if $k$ is not, in fact $\mathbb C$.

Thus, we can write $A$ as the category of representations of $\pi_1^{top} \to GL_n (\mathbb C) $

Any $l$-adic construction is going to be about continuous representations $\pi_1^{et} \to GL_n(\mathbb C)$ for some topology on $GL_n(\mathbb C)$.

To get an equivalence of categories in any kind of nice way, we clearly need every representation $\pi_1^{top} \to GL_n(\mathbb C)$ to extend to a continuous representation of $\pi_1^{et}$. So its image must lie in a compact subgroup. But $GL_n(\mathbb Q_l)$ has elements, like $\frac{1}{l} I $, that do not lie in any compact subgroup. I don't see any way to modify this construction to fix that bug.

For 3), you may find my answer here interesting.

One has big problems trying to make category $B$ work.

Standard Lefschetz principle arguments justify Lars' use of $\mathbb C$ even if $k$ is not, in fact $\mathbb C$.

Thus, we can write $A$ as the category of representations of $\pi_1^{top} \to GL_n (\mathbb C) $

Any $l$-adic construction is going to be about continuous representations $\pi_1^{et} \to GL_n(\mathbb C)$ for some topology on $GL_n(\mathbb C)$.

To get an equivalence of categories in any kind of nice way, we clearly need every representation $\pi_1^{top} \to GL_n(\mathbb C)$ to extend to a continuous representation of $\pi_1^{et}$. So its image must lie in a compact subgroup. But $GL_n(\mathbb Q_l)$ has elements, like $\frac{1}{l} I $, that do not lie in any compact subgroup. I don't see any way to modify this construction to fix that bug.

For 3), you may find my answer here interesting.