The Thomason model structure on the category of small categories is transferred from the Quillen model structure on simplicial sets along the right adjoint $Ex^2 \circ N$ (where $N$ is the nerve), i.e. weak equivalences and fibrations are created by this functor. It is known that this model structure is cofibrantly generated (by the left adjoint $c \circ Sd^2$ applied to the usual generators in $sSet$) and is Quillen equivalent to $sSet$ along the adjoint pair $(c \circ Sd^2, Ex^2 \circ N)$. Furthermore, $Cat$ has the structure of a category enriched, tensored, and cotensored over $sSet$ (one such structure is described in Rezk's note).

Is there such a structure on $Cat$ that satisfies Quillen's SM7 axiom to make the Thomason model structure into a simplicial model category?

Some sufficient conditions to check can be found in Lurie's Higher Algebra, Prop, but don't seem likely to be satisfied in this case.

Similarly, $Cat$ has two monoidal structures: the Cartesian one and the "funny" one.

Do either of the monoidal structures satisfy the pushout product axiom to make the Thomason model structure into a monoidal model category?

On the one hand, its homotopy category certainly is monoidal. On the other hand, the cofibrations don't seem to be super well understood (though, it is known that every cofibrant object is a poset), and I don't know much about the monoidal properties of the functors $Ex^2$ and $N$.

UPDATE: Tim's answer below has an attractive argument, but the comments lay out the ways it's wrong. First, Tim's argument fails between the two displayed diagrams. Secondly, even if the Cartesian structure did satisfy the pushout product axiom, the counterexample shows that one simplicially enriched structure on Cat (the one you get from applying the nerve functor to the Cartesian internal hom) fails the SM7 axiom. However, there does not seem to be a counterexample yet to the question of whether or not the Cartesian structure satisfies the pushout product axiom, and we still don't know if there exists a simplicial structure that satisfies the SM7 axiom. As far as I can tell, both questions in the current thread are still open.

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    $\begingroup$ $\operatorname{Ex^2}$ preserves products, as does the nerve. Both are right adjoints. Does this help? $\endgroup$ – Harry Gindi Nov 25 '18 at 23:19
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    $\begingroup$ The natural way to do this would be to have the given Quillen adjunction be an equivalence of $\mathrm{SSet}$-enriched model categories, which would amount to it preserving the simplicial enrichment. The usual way this would happen would be if left adjoint preserved finite products, which it does not-for instance, look at a square. So I'm skeptical whether this can be done. $\endgroup$ – Kevin Carlson Nov 26 '18 at 4:36
  • $\begingroup$ What is the "funny" monoidal structure? $\endgroup$ – Cihan Nov 29 '18 at 15:08
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    $\begingroup$ @Cihan ncatlab.org/nlab/show/funny+tensor+product $\endgroup$ – Tim Campion Dec 5 '18 at 20:46
  • $\begingroup$ I just want to record here a lesson I learned from Kevin Carlson in my now-deleted answer: $sd\dashv Ex : sSet \to sSet$ is an adjunction, but it's not a simplicial adjunction! This sinks the attempted proof I had. $\endgroup$ – Tim Campion Jan 12 at 2:49

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