> Let's say you need a short segment of DNA to randomly pop into existence

But do you? If elements A and B tend to form compound A-B, and compound A-B can attach a further set of A and B, forming B-A-B-A which under certain circumstances will split into A-B and A-B, you have basic reproduction, and perhaps the occasional presence of element C will expedite the process. Game of life. In my [very] layman's view of the world, given a billion years or two, some such infinitely simple process should be enough to get the ball rolling, as long as it can somehow produce an ever so slightly errorprone copy.