Very good post. The most probable theories of how living things arose from non-living things propose a very gradual process. It has already been demonstrated that amino acids and other organic molecules which occur importantly in living things can appear spontaneously under conditions which may approximate those existing in the early history of the Earth. A very crucial step in these theories is the occurrence of a self replicating molecule. As you quite correctly point out, natural selection has no meaning until there is reproduction.
Reproduction is one of the basic requirements for life, but there are others.
Nobody knows how many self replicating molecules are possible. DNA happens to be the one all terrestrial life is based on, but it may not be the only molecule that could lead to living organisms. There are several answers to this. One is that they probably do arise spontaneously but the environment is now so filled with highly developed living forms that these new arrivals in the contest don't have a chance. They get absorbed.