Firstly, so long as you plan to have horizontal exhaust/chimney-pipes: yes, you need the barrel. Also, yes the barrel acts as a radiator. The combination of the insulated internal-chimney (the heat-riser) and the radiating barrel are what create the strong "rockety" draft which pushes the air through the horizontal exhaust system. Here's how it happens: hot air rises up the heat-riser. As the air exits the top the heat-riser, it enters into the area between the barrel and the heat-riser. The barrel acts as a radiator and cools the hot air. As the air cools, it drops down and out the exhaust. This process creates a strong draft which pulls ('rockets') air in through the burn-tunnel and out through the exhaust pipes.
This is also why it is important to insulate the heat-riser. If the heat riser is made of merely brick or metal, it wouldn't be well insulated: heat would conduct through the brick or metal and would convect into the 'cooling-area' (between the heat-riser and barrel). That would screw-up the cooling process, and hence screw up the rockety-ness of the stove. And hence cause a lot of 'smoke-back' (backwards flow of smoke, into the room, rather than out the exhaust).
6" ducting is the standard 'small' size rocket mass heater. Anything smaller than that starts to create problems: while you can try to make the whole stove smaller and smaller and maintain proportions by adjusting variables accordingly, the air-friction across the interior surfaces of the pipes is considered to be a 'constant' (not a variable). So, due to that being a constant, there is a limit to how small the design can be scaled down, before approximate proportionality is lost.
For example, if you made a 4" duct system, the draft wouldn't be strong enough to push the air through a very long 4" pipe. So the exhaust pipe would have to be shorter-- but if the exhaust pipe is short, then the air exiting the exhaust is going to be hotter, meaning that the stove is less efficient (reaping less heat before the air goes outside). A remedy for this may be to make the heat-riser taller-- supposedly that creates a stronger draft. But my knowledge on this is limited. Look around the forum some more if you are interested.
8" is a pretty standard duct-size too. Should work well.