Pretty brief reply, but Pulsar uses timing belts because they give some slip to the system and it means it can go directly to the motors without the need for a clutch, because of the size and surface area of the weapon it has the torque to use this. carbide, other robots with a weapon that size, they need more torque to spin up and a smooth pulley would slip too much so chain is better, but it means they need a clutch to stop the motor dying from impacts.

Shaft size does matter yeah, most featherweights go for around 20mm thick shafts with as much support as possible. Deadshafts (stationary) systems normally have the bearings in the hub/pulleys either side of the weapon, advantage to using a dead shaft is that the shaft can be used to support the bulkheads and be a structural component in the robot.