Equation of continuity for stones

The answer to the first question is that the rate of flow involves not just the velocity and cross sectional area, but also density. With stones in a pipe, the area stays the same, so when the velocity rises, the density falls to compensate (the distance between stones get stretched vertically as their velocity rises). With a fluid, we often assume interparticle forces that maintain a fixed density, so as the velocity rises, the cross sectional area is what drops rather than the density. The interparticle forces, absent with stones, make that possible.

The answer to the second question is that the mass continuity equation assumes you have a steady state, i.e., the state of the fluid as a whole looks the same from moment to moment. You can't have that if you want to keep feeding in new fluid at the top yet claim it is dropped with zero speed-- that fluid has to come from somewhere, so must have some kind of motion. In the case of a water faucet, there is horizontal flow that is turning into vertical flow, so just analyze the flow along the pipe direction as the pipe turns from horizontal to vertical. There is never anywhere that has zero speed along the pipe once a steady state appears. And if you only look at the region where the flow is entirely vertical, there is always some nonzero vertical motion, even at the top of the downward flow.

1) About the rocks, you're ignoring the role of air. As they speed up, they separate, so air comes in to fill the spaces. With water, that can't happen (before drops form), so air pressure (and surface tension) squeezes the stream together.

2) About the water, you're ignoring the thickness $y_1$ of the slug of water, which has volume $A_1y_1$. After it has fallen and reached area $A_2$ it still has the same volume, so $y_2$ is larger.

First question: The equation of continuity is: d1*v1*a1=d2*v2*a2 where d1 and d2 are intial and final density, v1 and v2 are initial and final velocity and a1 and a2 is initial and final cross sectional area. The equation you are using is a special case of the above mentioned equation where the density remains constant. When we throw stones, their density is reduced. This is because the distance between them increases due to free fall. So the cross sectional area is not reduced and the equation is not violated as density is reduced. Second question: The initial velocity in the equation is the rate at which the fluid enters the tube, initial velocity is zero implies that the fluid is not entering the tube. Try to understand that the initial velocity here cannot be zero because there has to be some rate at which the fluid is entering the tube.