You posted this same video last month:
http://nofilmschool.com/2017/02/sound-design-foley-fight-club

The coil lives with the unit because the unit may have to move but the power supply will not.

A 20A cable can be coiled just fine without problems. Larger cable should be figure-eighted. Either can be left near the base of the light stand without problem (heat goes up). The issue in your picture looks like the result of pulling more load than the cable is rated for, although coiling an overloaded cable will only make heat dissipation worse.