Increasingly harder question, but a reference for the first would be ok:
Is the category of (symmetric?) monoidal categories closed for limits like products?
Is it true that the underlying category is the limit of the underlying categories? Would be ok if you can build a "free" monoidal category out of a category.
In case (1) is "no", along which diagrams you can take the limit? Seems like there are no problems in the product (just set everything component by component).