The article by Gepner-Groth-Nikolaus is the canonical reference for the tensor product of $E_\infty$-spaces. In the end it is quite a formal construction so there is not that much to say. A useful point of view that does not appear in loc. cit. is that this tensor product comes from the Lawvere theory of commutative monoids. To explain this, consider the $(2,1)$-category $\mathrm{Span}(\mathrm{Fin})$ whose objects are finite sets and whose morphisms are spans $I\leftarrow K\rightarrow J$. It has the following universal property: for any $\infty$-category $\mathcal C$ with finite products, there is an equivalence
$$
\mathrm{CMon}(\mathcal C) = \mathrm{Fun}^\times(\mathrm{Span}(\mathrm{Fin}),\mathcal C),
$$
where $\mathrm{Fun}^\times$ is the $\infty$-category of functors that preserve finite products. Since $\mathrm{Span}(\mathrm{Fin})$ is self-dual, this means that $E_\infty$-spaces are finite-product-preserving presheaves on $\mathrm{Span}(\mathrm{Fin})$:
$$
\mathrm{CMon}(\mathcal S) = \mathcal P_\Sigma(\mathrm{Span}(\mathrm{Fin})).
$$
This was first studied in the thesis of J. Cranch. From this perspective, the direct sum and tensor product are the Day convolutions of $\sqcup$ and $\times$ on $\mathrm{Span}(\mathrm{Fin})$ (here $\times$ means the usual product of finite sets, which is not the categorical product in $\mathrm{Span}(\mathrm{Fin})$; the latter is the same as the categorical coproduct, i.e., the disjoint union $\sqcup$). For example, $E_\infty$-semirings can be described as right-lax symmetric monoidal functors $(\mathrm{Span}(\mathrm{Fin}),\times)\to(\mathcal S,\times)$ that preserve finite products.
The unit. As Rune already explained, the unit for the tensor product of $E_\infty$-spaces is the free $E_\infty$-space on a point, that is the groupoid $\mathrm{Fin}^\simeq$ of finite sets with the $E_\infty$-structure given by disjoint union. This is equivalently the presheaf on $\mathrm{Span}(\mathrm{Fin})$ represented by the point, which is the unit for $\times$ on $\mathrm{Span}(\mathrm{Fin})$.
Here are a few examples I could think of. Let $E\in \mathrm{CMon}(\mathcal S)$.
Tensoring with a free $E_\infty$-space. Let $X\in\mathcal S$. Then
$$
\left(\coprod_{n\geq 0} (X^n)_{h\Sigma_n}\right) \otimes E = \operatorname{colim}_X E,
$$
where the colimit is taken in $\mathrm{CMon}(\mathcal S)$. This follows from the case $X=*$ using that $\otimes$ preserves colimits in each variable.
Tensoring with $\mathbb S$. Tensoring with the sphere spectrum $\mathbb S$ is the same as group-completing:
$$
\mathbb S\otimes E = E^\mathrm{gp}.
$$
For example, for a ring $R$,
$$
\mathbb S\otimes \mathrm{Proj}(R) = K(R).
$$
where $\mathrm{Proj}(R)$ is the groupoid of finitely generated projective $R$-modules, and $K(R)$ is the K-theory space.
Tensoring with $\mathrm{Fin}^\simeq[n^{-1}]$. Another localization of $\mathrm{CMon}(\mathcal S)$ is obtained by inverting integers (or rather, finite sets). The inclusion of the full subcategory of $E_\infty$-spaces on which multiplication by $n$ is invertible has a left adjoint $E\mapsto E[n^{-1}]$, which is equivalent to tensoring with $\mathrm{Fin}^\simeq[n^{-1}]$. But unlike in the cases of either abelian monoids or spectra, $\mathrm{Fin}^\simeq[n^{-1}]$ is not just the sequential colimit of multiplication by $n$ maps; it is obtained from the latter by killing suitable perfect subgroups of its fundamental groups, in the sense of Quillen's plus construction, to ensure that $n$ acts invertibly.
Tensoring with $\mathbb N$. Let $\mathrm{FFree}_{\mathbb N}$ be the 1-category of finite free $\mathbb N$-modules. There is a functor
$$
\mathrm{Span}(\mathrm{Fin}) \to \mathrm{FFree}_{\mathbb N}
$$
sending a finite set $I$ to $\mathbb N^I$, inducing an adjunction
$$
\mathrm{CMon}(\mathcal S) = \mathcal P_\Sigma(\mathrm{Span}(\mathrm{Fin})) \stackrel{\mathrm{str}}\rightleftarrows \mathcal P_\Sigma(\mathrm{FFree}_{\mathbb N}).
$$
Objects in the right-hand side are sometimes called strictly commutative monoids (the group-complete ones are connective $H\mathbb Z$-module spectra). Tensoring with $\mathbb N$ amounts to strictifying a commutative monoid in this sense:
$$
\mathbb N\otimes E = E^\mathrm{str}.
$$
Unlike $\mathbb S$, $\mathbb N$ is not an idempotent semiring, that is, strictifying is not a localization. Indeed, $\mathbb N\otimes\mathbb N$ is an $E_\infty$-space whose group completion is the "integral dual Steenrod algebra".
Tensoring with $\mathrm{Vect}_\mathbb{C}^\simeq$. Let $\mathrm{Vect}_\mathbb{C}^\simeq=\coprod_{n\geq 0} BU(n)$, where $U(n)$ is regarded as an $\infty$-group (despite the notation, this is not really the core of an $\infty$-category of vector spaces). This is an $E_\infty$-space whose group completion is $\mathrm{ku}$. There is a related $\infty$-category $2\mathrm{Vect}_{\mathbb C}$ whose objects are finite sets and whose morphisms are matrices of complex vector spaces. As in the previous example we get an adjunction
$$
\mathrm{CMon}(\mathcal S) = \mathcal P_\Sigma(\mathrm{Span}(\mathrm{Fin})) \rightleftarrows \mathcal P_\Sigma(2\mathrm{Vect}_{\mathbb C}).
$$
An object in the right-hand side is roughly speaking a commutative monoid such that $U(n)$ acts on the multiplication by $n$ map in a coherent way. Tensoring with $\mathrm{Vect}_\mathbb{C}^\simeq$ gives the free commutative monoid with such structure.
For $E_\infty$ spaces, homotopy-theoretically there is a functor $L: \mathcal{S} \to E_\infty \mathcal{S}$ with a right adjoint $R$. The only property on this list that really needs replacing on this list is property (5): the unit
$$
X \to RL(X)
$$
should be homotopy equivalent to the natural inclusion
$$
X \to Free_{E_\infty}(X) \simeq \coprod_{k \geq 0} E \Sigma_k \times_{\Sigma_k} (X^k)
$$
into the free $E_\infty$-space on $X$. (Yes, yes, possibly a version with basepoints, I know)
I believe that all three of the models of $E_\infty$ spaces that you list (commutative monoids in $*$-modules, $\Gamma$-spaces, commutative $\mathcal{I}$-space monoids) satisfy properties (1)-(4) and fail the analogue of property (5), each due to an issue about whether an input to an adjunction is cofibrant/fibrant. For $\Gamma$-spaces, for example, the map $X \to RL(X)$ only adds a disjoint basepoint. Perhaps someone with more experience with the other models would be able to fill in those stories better.
Best Answer
Yes, for the same reason. Let me sketch a proof.
1- $QS^0\otimes X$ is group-complete. Indeed, its $\pi_0$ is $\mathbb Z\otimes \pi_0(X)$, and that's a group for the usual reasons. Another way to prove it is to prove that the shear map for $X\otimes Y$ is (the shear map of $X)\otimes Y$, which can be seen by noting that $\otimes$ commutes with coproducts and hence finite products in each variable.
2- There is a natural transformation $X\to QS^0\otimes X$ given by tensoring $\mathbb F\to QS^0$ by $X$, and this induces a natural transformation $X^{gp}\to QS^0\otimes X$.
3- Both sides commute with colimits (a colimit of grouplike $E_\infty$-spaces is grouplike so I don't have to worry about whether I'm talking about colimits in monoids or grouplike monoids), therefore to check that this map is an equivalence, it suffices to do so for $X= \mathbb F$, and for that one it is a tautology.
Another way to phrase this is to use the following sequence of natural equivalences (and using point 1- for the last one):
$X^{gp} = QS^0\otimes_{QS^0} X^{gp} = (QS^0\otimes_\mathbb F X)^{gp}= QS^0\otimes X$
The second natural equivalence comes from the fact that group completion is symmetric monoidal, and $(QS^0)^{gp}\simeq QS^0$.