I have the following questions on monoidal model structure(s) for the motivic stable homotopy category $SH(k)$ (where $k$ is a field); certainly, I am also interested in general statements concerning these matters.

Which "models" for $SH(k)$ are monoidal model categories (with the extra monoid axiom fulfilled) such that the unit object (sphere spectrum) is cofibrant?

For $MGl'$ being a cofibrant ring spectrum in this category that is weakly equivalent to the Voevodsky algebraic cobordism spectrum $MGl$ (over $k$) I would like to have a criterion that ensures that a cohomological functor from $SH(k)$ (into abelian groups) factors through the (triangulated) homotopy category $D^{MGl}$ of highly structured $MGl'$-modules. Any suggestions?

3.[My own idea concerning 2]. The spectrum $MGl$ possesses a universality property (see https://projecteuclid.org/euclid.hha/1251811074) that ensures that various forms of K-theory are represented by spectra that are algebras over $MGl$ in $SH(k)$. To prove that these "K-theories" factor through $D^{MGl}$ I would like to prove that certain "replacements" of these K-theory spectra are highly structured $MGl'$-modules. It seems reasonable to consider the model structure for ring spectra in the model for $SH(k)$ that is provided by http://www.math.uni-bonn.de/~schwede/AlgebrasModules.pdf Yet to use this model structure I probably need a calculation of morphisms in the corresponding homotopy category; is anything known about this?

Proposition 1.10 of Hovey's unpublished http://mhovey.web.wesleyan.edu/papers/mon-mod.pdf says that for a left proper cellular monoidal model category $C$ the category of modules over a cofibrant monoid object $R$ of $C$ is (cellular and) left proper also. Does a (published) proof of this fact or something similar exist? Cf. the discussion at Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology)

I believe that the category of $R$-modules is a $C$-module category in the sense of Definition 4.2.18 of Hovey's book. Does there exist a reference for this statement?