Is the Galilean Cannon a demonstration of Newton's Third Law? How?

By the third law, you know that the force from the collision is identical on each component. But forces are not simple to calculate in collisions.

The impulse (total change in momentum) is straightforward, but the time that this takes is unknown. When the materials are very rigid, the collision time is very short and the forces are very large.

Further, in the cannon, it's not necessary to assume that all the forces act simultaneously. Instead, you can assume that each object collides sequentially (starting from the bottom). In that case, the forces that arise are only due to the acceleration of the colliding objects, not due to additional forces from neighbors.

When done this way, you can assume that each pair of objects approach and leave the collision at the same speed, when viewed from a frame where their center of mass is at rest. Knowing the initial speed and the masses, this can be calculated.