Why does this flame travel (in a circular path, multiple times)
A flame is a dynamic phenomenon, using up fuel and oxygen in places and driven by convection and diffusion. Flames are sensitive to factors such as external sound, and can be used to measure frequency and waveform with a Rubens tube.

Under micro-gravity, flames burn quite differently, due to the lack of convection.

So though not a specific answer to your questions, I assume that oxygen is being used up rapidly in one spot, and the flame front moves along as convection brings in fresh oxygen. You could test this hypothesis by piping in just a little air in one spot on the "ring of fire" to see if it stabilizes the flame there.

Thanks for sharing that interesting video!