What is the cross-product of the null set with another set?

Yes, you're correct: For all sets $B$, if $A = \varnothing$, then $$A \times B = \varnothing \times B = B\times A = B\times \varnothing = \varnothing$$ and for the reason you argue: there exist no ordered pairs in $\varnothing \times B$, by the definition of the Cartesian Product.

Yes, you are correct.

The cartesian product is defined as $$A \times B = \{(a,b) \mid a\in A, b\in B\}.$$

In the case that one or both of the sets $A$ and $B$ are empty, there is no single index pair $(a,b)$ such that $a\in A$ and $b\in B$, which means $A\times B = \emptyset$.