These results follow from the Cartan-Leray spectral sequence, which for a regular covering map $X\to X/W$ and a commutative ring $k$ of coefficients has $$ E_2^{p,q}=H^p(W,H^q(X;k)) $$ (cohomology of the group $W$ with coefficients in the $kW$-module $H^\ast(X,k)$) and converges to a graded group associated to $H^\ast(X/W)$. A reference is Ken Brown's "Cohomology of groups", section VII.7.
In case the group $W$ is finite, if $|W|$ is invertible in $k$ then $H^p(W;H^q(X;k))=0$ for all $q$ and all $p>0$ (see Brown, Corollary III.10.2). In particular this is true if $k=\mathbb{Q}$. Thus the spectral sequence is concentrated in the $0$ column and therefore collapses, giving $H^\ast(X/W)\cong H^0(W;H^\ast(X;k))$. Since $H^0(W;M)=M^W$ for any group $W$ and any $W$-module $M$, this gives the stated results. So you were exactly right in your first paragraph!
More generally, the same collapse happens for the Serre spectral sequence of the fibration $X\to X_W\to BW$, which has $$ E_2^{p,q}=H^p(BW;H^q(X))\cong H^p(W;H^q(X)), $$ giving the isomorphism $H^\ast_W(X)\cong H^\ast(X)^W$ you mentioned.