I don't think self-organization is equivalent to self-design. As I understand it, the former is when a dynamic process is able to maintain or reduce its local entropy at the expense of its environment. The latter implies intent and choice.
That is correct as I understand it also. However, randomly generated cellular automata have a 1/256 chance (or better) of being a Universal Turing Machine, and thus can self-reproduce and can compute any computable function.
For that matter, there was a self-organized nuclear reactor in Africal some years ago.
There is a tendency to equate self-organizing complexity with self-replication. As a metaphor, if the cells were only self-replicating, a fetus would look like a tumor.
Also, the emphasis in self-organizing complexity is on the "self". A snowflake, for instance, is not an example of self-organizing complexity because it is made structurally complex by external forces. For more: Syntactic Autonomy: Or Why There is no Autonomy Without Symbols and how Self-Organizing Systems Might Evolve Them