In the text's impulse-related equations, $J=-n\Delta p$ and $F_{\rm avg} = -\frac{n}{\Delta t} m \Delta v$, when would we use them?

I think you're referring to the expressions on p. 228 of the text. This is really just considering the 1D case of multiple collisions, and it's just expressing that the total impulse and average force on an object that's colliding with $n$ projectiles are the sums of those from each individual collision.

This is not one of those very general expressions you'll use in many contexts, but rather an application of more basic ideas.