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Let us try to figure out what's happening on discrete rings, where $E_2=E_\infty$. The category $\mathrm{LMod}^{(2)}$ is, as you surmised, the category of pairs $(A,B)$ where $A$ is a commutative alge …

Let $\mathcal{C}$ be an $\infty$-operad. Then Lurie in Higher Algebra, section 3.3.3 constructs a family of $\infty$-operads $$\operatorname{Mod}(\mathcal{C})^\otimes\to \operatorname{Fin}_\ast \times …

In fact what you need is that your ∞-category is additive (i.e. that it has direct sums and that the canonical commutative monoid structure on the mapping spaces is group-like). All stable categories …

(2) is true (and so (1) is false). To see it, note that every horn $\Lambda^n_i\to S$ to a constant simplicial set must be constant, and so it can be filled by the constant horn $\Delta^n\to S$. Equi …

Let me add a short observation to Dylan's fantastic answer. There is indeed a more concrete construction of the symmetric monoidal structure on the $\infty$-category of spectra: it is the localized Da …