When is the etale cohomology of $\mathrm{Sym}^n(X)$ isomorphic to the $\Sigma_n$-invariants in the étale cohomology of $X^n$?

One can give a spectral-sequence free argument. Let $X$ be an algebraic variety and $G$ a finite group acting on $X$, acting freely on a dense open subset. Let us say that $X$ is quasi-projective so that $X/G$ exists as a scheme (rather than an algebraic space), but this is not essential. We have $\pi \colon X \to X/G$ which induces a pullback map $\pi^\ast \colon H^\ast(X/G) \to H^\ast(X)$ as well as a trace map $\pi_\ast \colon H^\ast(X) \to H^\ast(X/G)$. The trace map exists in great generality; if $X$ and $X/G$ are both smooth the trace can be defined simply as the Poincaré dual of the pushforward map in étale homology. This is not true in your case but your spaces are still rational homology manifolds so that Poincaré duality holds with $\mathbb Q_\ell$-coefficients. In any case, what makes it all work is that $\pi_\ast \pi^\ast$ is multiplication by $\vert G\vert$ on $H^\ast(X/G)$, and that $\pi^\ast \pi_\ast$ is the map $x \mapsto \sum_{g \in G} g \cdot x$ on $H^\ast(X)$. In particular, if $\vert G\vert$ is invertible in the coefficients of the cohomology then $\pi_\ast \pi^\ast$ is invertible and $\pi^\ast \pi_\ast$ is projection onto the $G$-invariants.

Unfortunately I couldn't tell you a citable reference off hand. If I were to use this in a paper I wouldn't give a reference for this - sorry, I know that's not a very helpful thing to say.

The following is a particular case of (SGA 4.3, XVII Th. 5.5.21) : Let $X$ be a quasi-projective scheme over an algebraically closed field $k$. Then for any $n \geq 0$ and any $r \geq 1$ we have $$ R \Gamma_c(\mathrm{Sym}_{k}^n(X), \mathbb{Z}/r\mathbb{Z}) = L \Gamma^n ( R\Gamma_c(X, \mathbb{Z}/r\mathbb{Z})). $$ The functor $L \Gamma^n$ is the left derived functor of the non-additive functor $\Gamma^n$, which coincides on flat modules with the ``symmetric tensor'' functor. The $R \Gamma_c$ denotes higher direct image with compact supports.

(SGA 4.3, XVII Th. 5.5.21) gives a more general statement, in a relative situation, with more general coefficients.