Typically, the solid oxides would be heated in a reducing (H2) environment, forming water vapor and liquid metal (or suboxide).

You could get that reducing gas from hydrolysis, and solar, but it's not energy cheap (~50% efficiency).

This is a good question. Metal oxides tend to decompose at high temperatures, when exposed to reducing agents, under vacuum, or with electricity. I can’t immediately find good sources about which of these apply to gallium and would be most energetically favorable.

But regardless it is certainly possible to regenerate it using only electricity possibly with intermediate but renewable substances (water, sodium chloride, etc)

Probably O2 is a problem because it will oxidize gallium as well.

it has a very high melting point (1900 ˚C) and their experiment capped out at 200 °C so presumably it crystalizes and can be scooped out as well?