What is required for a monoidal category to have products/coproducts? What is required for a monoidal category to have products/coproducts?
If it helps the particular category I am interested in also has zero morphisms. 
 A: A category with products is always monoidal, where the tensor product coincides with the product.
On the other way round (which is what I thought you're asking), a category that has all finite products, including the empty product, is called a cartesian category. So, the question is "What does it take for a monoidal category to be cartesian?". It turns out that a cartesian category is a monoidal category that is also


*

*symmetric$^1$: $\gamma_{B,A}\circ\gamma_{A,B}=\mathbf{1}_{A\otimes B}$, where $\gamma_{A,B}:A\otimes B\overset{\sim}{\to}B\otimes A$ is the braiding isomorphism

*and the monoidal structure of the category comes from the product, meaning $\otimes=\times$.


If the category is also closed (for any fixed object $A$, functor $(-)\otimes A$ has a right adjoint) then the category is cartesian closed.
I suggest you take a look at the book "Category theory for Computing Science", from Barr and Wells.
1: Note that for a monoidal category to be symmetric, it needs to be braided, too.
