The answer to both questions is no, since a monoidal category can have several non-equivalent symmetric structures. For instance, the category of $\mathbb{Z} / 2\mathbb{Z}$-graded vector spaces can be given the "usual" symmetry, 
$$c(x \otimes y) = y \otimes x,$$
or the "super" symmetry,
$$c(x \otimes y) = (-1)^{\lvert x \rvert \lvert y \rvert} y \otimes x.$$
Thus, the identity functor between the category of $\mathbb{Z} / 2\mathbb{Z}$-graded vector spaces with the "usual" symmetry and the category of $\mathbb{Z} / 2\mathbb{Z}$-graded vector spaces with the "super" symmetry is a monoidal equivalence, but not symmetric monoidal.