The "heat" IS the radiation. So, yes.

Someone correct me if I'm missing some nuance here, but heat doesn't get transferred directly through space because heat is vibrating molecules and space is a vacuum. The sun radiates (speed O' light). A lot of that radiation just reflects off the earth (or we wouldn't be able to see it), but a lot of it gets absorbed. THAT's when it's converted into heat energy. It's also why the greenhouse effect is a global phenomena: light energy comes in across the vacuum relatively easily, turns to heat on Earth instead of being reflected, heat energy cannot escape as easily as light energy.

When there is a total solar eclipse, the temperature does drop dramatically. But it might not be detectable on the other side right away for sure.

Yes, we have conduction, convection, and radiative heat transfer. Vacuum insulates the first two, it's the light from the sun that heats us up