The answer to 2) is yes for cofibrant operads, see http://arxiv.org/abs/1102.1311 by Fiedorowicz and Vogt. The answer to 1) is no; the Boardman-Vogt tensor product does not interact well with cofibrations.

EDIT: As an answer to Chris' comment, here is a counterexample to 2) in the setting of colored operads, although you can adapt it to the one-color case as well. In fact, the example just concerns simplicial categories (i.e. simplicial operads with only unary operations), where the tensor product coincides with the Cartesian product of simplicial categories. Therefore it also shows why the Bergner model structure on simplicial categories is not Cartesian.

Write $I$ for the category with objects 0 and 1, with one non-trivial morphism from 0 to 1, and write $\partial I$ for the disjoint union of 0 and 1 (with just their identity morphisms). Clearly $\partial I \rightarrow I$ is a cofibration. However, the pushout-product $\partial I \times I \cup I \times \partial I \rightarrow I \times I$ is not a cofibration; it's not even a monomorphism. One the left-hand side, the (discrete) simplicial set of morphisms from $(0,0)$ to $(1,1)$ consists of two points, on the right-hand side there is only one.

If you tweak this example to apply to the one-object case (and identifying categories with one object with monoids), you run into the morphism from the free monoid on two generators to the free commutative monoid on two generators, which again is not a monomorphism.

By the way, this is not the only issue: it is also possible to cook up counterexamples to the pushout-product axiom for the Boardman-Vogt tensor product by playing around with nullary operations, using an Eckmann-Hilton style argument.

The answer to 2) is yes for cofibrant operads, see http://arxiv.org/abs/1102.1311 by Fiedorowicz and Vogt. The answer to 1) is no; the Boardman-Vogt tensor product does not interact well with cofibrations.

EDIT: As an answer to Chris' comment, here is a counterexample to 2) in the setting of colored operads, although you can adapt it to the one-color case as well. In fact, the example just concerns simplicial categories (i.e. simplicial operads with only unary operations), where the tensor product coincides with the Cartesian product of simplicial categories. Therefore it also shows why the Bergner model structure on simplicial categories is not Cartesian.

Write $I$ for the category with objects 0 and 1, with one non-trivial morphism from 0 to 1, and write $\partial I$ for the disjoint union of 0 and 1 (with just their identity morphisms). Clearly $\partial I \rightarrow I$ is a cofibration. However, the pushout-product $\partial I \times I \cup I \times \partial I \rightarrow I \times I$ is not a cofibration; it's not even a monomorphism. On the left-hand side, the (discrete) simplicial set of morphisms from $(0,0)$ to $(1,1)$ consists of two points, on the right-hand side there is only one.

If you tweak this example to apply to the one-object case (and identifying categories with one object with monoids), you run into the morphism from the free monoid on two generators to the free commutative monoid on two generators, which again is not a monomorphism.

By the way, this is not the only issue: it is also possible to cook up counterexamples to the pushout-product axiom for the Boardman-Vogt tensor product by playing around with nullary operations, using an Eckmann-Hilton style argument.

The answer to 2) is yes for cofibrant operads, see http://arxiv.org/abs/1102.1311 by Fiedorowicz and Vogt. The answer to 1) is no; the Boardman-Vogt tensor product does not interact well with cofibrations.

The answer to 2) is yes for cofibrant operads, see http://arxiv.org/abs/1102.1311 by Fiedorowicz and Vogt. The answer to 1) is no; the Boardman-Vogt tensor product does not interact well with cofibrations.

EDIT: As an answer to Chris' comment, here is a counterexample to 2) in the setting of colored operads, although you can adapt it to the one-color case as well. In fact, the example just concerns simplicial categories (i.e. simplicial operads with only unary operations), where the tensor product coincides with the Cartesian product of simplicial categories. Therefore it also shows why the Bergner model structure on simplicial categories is not Cartesian.

Write $I$ for the category with objects 0 and 1, with one non-trivial morphism from 0 to 1, and write $\partial I$ for the disjoint union of 0 and 1 (with just their identity morphisms). Clearly $\partial I \rightarrow I$ is a cofibration. However, the pushout-product $\partial I \times I \cup I \times \partial I \rightarrow I \times I$ is not a cofibration; it's not even a monomorphism. One the left-hand side, the (discrete) simplicial set of morphisms from $(0,0)$ to $(1,1)$ consists of two points, on the right-hand side there is only one.

If you tweak this example to apply to the one-object case (and identifying categories with one object with monoids), you run into the morphism from the free monoid on two generators to the free commutative monoid on two generators, which again is not a monomorphism.

By the way, this is not the only issue: it is also possible to cook up counterexamples to the pushout-product axiom for the Boardman-Vogt tensor product by playing around with nullary operations, using an Eckmann-Hilton style argument.