Ring structures on algebraic K-theory spectrum, and its non-connective counterpart

Have you looked at Chapter VI of EKMM? It's all about the algebraic K-theory spectrum. The intro says it was part of Mandell's PhD thesis. Indeed, Theorem 6.1 in that chapter seems to answer your (1) and (2). It says the algebraic K-theory spectrum KR is equivalent to an $E_\infty$ ring spectrum.

Also, $E_\infty$ does not depend on the model you use for spectra. You probably know Schwede's result that the stable homotopy category is rigid. It's also monoidally rigid (I think this was proven by Schwede and Shipley). Furthermore, because $E_\infty$ is encoded by a $\Sigma$-cofibrant operad, any monoidal Quillen equivalence lifts to a Quillen equivalence of $E_\infty$-algebras. See the Schwede-Shipley paper on monoidal equivalences.

Proposition 5.9 in [Blumberg, Gepner, Tabuada. Uniqueness of the multiplicative cyclotomic trace. arXiv:1103.3923] shows that both connective and nonconnective K-theory define lax symmetric monoidal functors (on the infinity-category of stable idempotent-complete infinity-categories). It follows that if you apply either variant of K-theory to a symmetric monoidal stable idempotent-complete infinity-category, you end up with an $E_\infty$-ring spectrum. Of course this applies to the infinity-category of perfect complexes on a commutative ring or $E_\infty$-ring spectrum.