How can the solar panel itself radiate heat when it's being heated up generating supplying power? Looking at pictures of the ISS there's radiators that look like they're there specifically to cool the solar panels.

And even if viable, why would you just not cool using air down on earth? Water is used for cooling because it increases effectiveness significantly, but even a closed loop system with simple dry air heat exchangers is quite a lot more effective than radiative cooling

You take the amount of energy absorbed by the solar panels and subtract the amount they radiate. Most things in physics are linear systems that work like this.