Pseudo-random, 'swaying', motion appears to help induce a flow state, in that it 'captures' the movements of an activity and entrains them to an underlying rhythm of activity. Time between activity-related movements is reduced, and error-recovery time is also reduced. (again, this is from first-hand observation, not published results).
The following is the relevant explanatory section of the 2nd question's answer:
Seen from this perspective, cyclical movements are the norm for almost any animal, whereas short-duration, "single-use" movements like typing or playing the piano are rather unusual. It could be the case that if the motor cortex (or even the basic movements encoded in the spinal cord) is inherently tuned to modeling cyclic movements, then adding some continuous motion could help the motor cortex capture the intended typing or playing movements as part of the larger, continuous movement.
The last question I have is if the frequency of the periodic movements, which in general follow a 'figure-8' track, could be related to the frequency of other cycles in the body? For example, could the frequency of motion reflect the active frequency of the brain, e.g. EEG waves? Or could it reflect a different neural/physiological cycle or state?
Naturally this is a tricky question: on one hand, the activity itself serves to entrain the neural system to a certain periodicity. On the other hand, if the rhythmic movement activity is allowed to 'float' in frequency organically, does its characteristics reflect anything about the initial state of the neural system?
Does the frequency of movement give clues to other periodic cycles, such as EEG waves?
Does said movement give any other insight into the brain/body state?