I guess I don't understand. How is the divergence theorem equivalent to Gauss' law? It seems like Gauss' integral law is a consequence of the divergence theorem, which is something you prove in Real Analysis class

This is a proof if you start with Maxwell's equations and the divergence theorem being given. What is your starting point then? Also, are you just downvoting answers you don't like, even if they aren't wrong?

This may be of interest to you: businessinsider.com/fastest-object-robert-brownlee-2016-2

Have you tried other Gaussian surfaces besides a cylinder?

I edited it. The dot product is the formula $\vec{F} \cdot \vec{d}$. If you haven't seen that, just pay attention to the part with the cosine (they are equivalent). The formula for work goes to $0$ when the force is perpendicular to the displacement ($cos(90^\circ)=0$). In your examples, gravity isn't always perpendicular to the displacement, so it can still contribute.

The work mentioned here goes to zero when the angle between the force and displacement is $90^\circ$ (assuming there is a nonzero force and displacement). In both cases you mentioned, the gravitational force is not always perpendicular to the displacement.