First, who appointed the author judge over what rewards are rightful and what are not?

They have PhDs... Therefore they think themselves appropriate judges for that.

the author's logic, he himself belongs to the commons, because the material and energy needed to produce his own being did not come from himself.

Beautiful. I shall invoke slavery now... Your body is not even your molecules, bitch!_

which case the answer to "Who owns the Sun" is simply "Whoever has the means to exclude its use."

I agree that it's (almost entirely?) all excludability, that is a technical and economic consideration, not a moral or political one