I move the carriage around myself all the time and I’m out in the weather right with them. They have shoes that protect them from the pavement and heat conduction doesn’t work that way anyway. If you knew anything about horses, you would know that they don’t have any nerves in most of their hoof; only in the frog which is covered in thick skin.
What do you think horses who live in the wild in the desert do? I guarantee the sand is hotter than the pavement, and it permeates their hooves more than pavement does because the hoof actually sinks into the sand. And no, there is no shade in certain parts of the desert for those horses to go to.
As for the bit, people make such a big deal out of nothing with that. Horses have a space in their mouth between their grazing teeth and their molars which is where the bit sits comfortably. It simply communicates different levels of pressure to guide the horse where to go. In the wild, they bite and kick each other to tell the herd where to go. Our method is much tamer than that.