There are a lot of questions here about Cormac McCarthy's The Road. I find question nine the most interesting: "Why do you think the author did not provide names for his characters?" I hope my discussion of this question will give you some insight and jumping off points for the rest.
I see a few reasons that McCarthy may have left his two protagonists (and everyone else for that matter) nameless. The first reason is that it gives these characters an "everyman"-like quality. Without names, the man and his son could be anybody. They could be your neighbors, your friends, your father and brother, you, me. This helps us as an audience to connect with their plight. It grounds the sensational circumstances of the apocalypse in something we can relate to. The characters are anonymous and therefore ubiquitous.
Another possibility for their namelessness is that it enhances the horror of their circumstances. We equate names with identity. Our names are as deeply entrenched in our senses of self as any other identifying characteristic: hair color, eye color, gender, age, etc. By denying these characters names, McCarthy strips them of their identities and they become something less than human. This plays elegantly into one of the major themes of the novel, questioning what it means to be human.
One last reason that McCarthy may have chosen to leave the protagonists unnamed is that doing so contributes to the monotonous mood of The Road. Many readers find this text difficult, not because of the language or the thematic elements, but because it is just so slow and, some would say, dull. This, however, seems to be a very intentional construction. The monotony of the struggle, punctuated by brief moments of tension, can be seen as a commentary on the way we live our lives. We walk down the road, one foot in front of the other, with very little happening. Sometimes we think about the past, sometimes we have moments of crisis, and in the end we either get where we're going, or we don't. McCarthy effects this criticism through repetition of images, actions, and even words. By not naming the boy and his father, he can only give them descriptors (man, father, boy, son) that repeat over and over again. This helps enhance the mood and atmosphere McCarthy is attempting to achieve.