Thanks, I was not aware of that. I'll take a look if and when I have time.

Akhil just taught me by email how to get a comment prompt. I apologize for my stupidity and will try not to err that way again. I don't mind a few negative votes. I have tried to give a mathematically convincing comment How to Answer re smash products of spectra. I agree with Tyler about consideration. While this is way off topic, I urge readers to look up the user list. You will see a remarkable paucity of women. I've talked with many, and I have often gotten the response that they dislike the general tone. Way too much showing off, too little helpful teaching.

To continue, as I've said in an answer to another question, I know many fine mathematicians who have given up participation on math overflow. I won't repeat the reasons. The vast majority of participants are novices (look up users) and are not greatly helped by the most sophisticated answer to the simplest of questions. A full mathematical answer to the question at hand depends on the categorical location of the smash product. Going from the category of (good) spaces (where the function space adjunction is a reasonable characterization) to the quasicategory of based spaces is a leap.

Apologies about answer vs comment. I don't spend much time on this and sometimes do not know how to comment (don't see a prompt). I write comments or answers when I want to try to teach something or when I see something that to my mind seriously misrepresents mathematics in a field I'm expert in. Many answers/comments seem seriously deficient to me. For example, it is not remotely a characterization of the symmetric monoidal smash product functor on spectra that "you do what you expect on (stable) cells and then glue up". That will not do and misrepresents the actual mathematics involved.

Akhil, no problem. For others, Akhil and I have corresponded off this toy and are on the same wavelength. Kevin, Akhil really didn't answer the question either. A symmetric monoidal infinity category is not the same animal as a symmetric monoidal plain category. A smash product in infinity category land does not answer a question about smash products as intended in the question. Incidentally, for the same reason, I do not believe you can define "the" smash product of spectra (as in Akhil's amendation) that way. What you get is only weakly analogous to tensor products.

Mark is right. Your category is connected, so its classifying space would be the zeroth space of the spectrum you would get if it were permutative. Then the fundamental group G would of course be commutative. Ignoring the notational problem with your definition of $h^n(X)$ ($\Sigma S^n$ is just $S^{n+1}$ and the difference between spaces and spectra is fudged) the spectrum you get when $G$ is abelian is just the suspension of the Eilenberg-Maclane spectrum $HG$, so it represents ordinary cohomology with coefficients in $G$. But you should learn the basics first, ask questions later.

$G$-CW isn't too restrictive (includes all smooth $G$-manifolds when $G$ is compact Lie). But here is an argument for weak equivalence without that assumption. A comparison of the Borel bundles for $X$ and $Y$ ($X\to EG\times_G X \to BG$) shows that the map of Borel constructions is a weak equivalence. The reduced Borel construction is the pushout of the maps $BG \to \ast$ and $BG \to EG\times_G X$. The latter map is a cofibration, so the gluing lemma for weak equivalences gives that the map of reduced Borel constructions is a weak equivalence.

Gut feelings of my collaborators and myself. But also the technical point that we doubted that the map $$(E\Sigma_j)_+ \wedge_{\Sigma_j} X^{(j)}\longrightarrow X^{(j)}/\Sigma_j$$ is an equivalence for cofibrant $W$-spaces $X$.

I hadn't thought about that, since to me the answer to the second question makes the first question uninteresting. With the obvious definitions, if you had a model structure then there would be a Quillen adjunction to commutative symmetric or orthogonal spectra that is not a Quillen equivalence. It seems more likely to me that you just don't get a model structure. But it is murky. Unclear what kind of prolongation you get from commutative orthogonal spectra when you know they can't give you what they ought to give you.