A brolly box is essentially a shoot-through umbrella. It is, however, more efficient than an umbrella because the back is closed. The inside of the closed back is either white or silver so that it redirects any light hitting it back through the front. Being essentially a shoot-through umbrella, it throws light in a wide, generally undirected pattern. If you've ever used a shoot-through umbrella in conjunction with the standard (generally a 7") reflector that comes with most heads then you've seen what a brolly box gives you.
Umbrellas, especially when used as a bounce, actually throw pretty even lighting over a broad area. The Westcott Master's Brush umbrella is an exception. It is designed to fall off rapidly from center to edge, allowing you more control over where you project your light.
The differences between the Chimera Super Pro and Super Pro Shallow boxes are subtle. Chimera will tell you that the Shallow boxes sacrifice a little bit of light quality and evenness in order to make a shallower box - useful in tight locations. Fundamentally, if you look at the profile of a regular softbox, the construction forms a parabolic shape. That parabolic shape is what contributes greatly to the efficiency, evenness, and softness of the box. The shallow box compromises that parabolic shape. All of that being said, you'd most likely have to shoot with the two boxes side-by-side in order to tell the difference.
The biggest difference between an octagonal box and a rectangular box of similar size is the shape of the light source. You get round or square highlights on reflective surfaces and round or square catch lights in a person's eyes. From a surface area standpoint, a 5' octagonal box (~19.5 square feet) sits between a 3'x4' rectangular box (12 sq ft.) and a 4'x6' rectangular box (24 sq ft.). I wouldn't say that one type is inherently advantageous over the other.
People who do stupid things with dangerous substances often die! -me