How can Stephanie's "stripped down" artifical mind possess this theoretical "genetic morality" when it is presupposed that her full artificial mind does not possess it in the first place? How can the part possess a feature that the whole does not?
We never hear that her mind does not posses the "genetic morality". The way I understood the conflict was like this:
Intelligences that evolve purely within a machine, meaning they do not come from a person, would not posses any morality. When that happens, we get Skynet.
People who's minds have been uploaded to a machine will at some point evolve beyond their current mental state. Minds evolve much faster than bodies do. Different stimulus on a mind forces it to react in different ways. The best example we see is that Stephanie's mind has evolved well past her 14 year old state.
Now, the fear here is that a person's mind can evolve so far away from its original humanity that it loses its morality, and we get Skynet again.
That is why any mind that is deemed to be post-human is destroyed, out of fear. (I'm not justifying that position, just explaining it as I understood it from the story).
Now, Stephanie's mind has been in a computer for the longest time ever. She fell between the chairs. She is living(?) proof that a post-human mind does NOT (always) lose its morality. So there is no fear of Skynet from her.
And yet, there could be future Skynet from others. What she does is to introduce genetic morality into the internet. And we all know that stuff on the internet spreads like wildfire. So, any intelligence evolving after that point (whether it started off as human or not) will encounter this morality and use it. Thus preventing Skynet and allowing the humans to trust the machines.
If it is supposed to somehow "evolve" after her mind is "stripped down":
1) why couldn't it evolve in situ in her own mind?
2) what forces in a virtual world would shape this evolution?
Stephanie's mind must be stripped down, because it cannot fit in its entirety in the public system. She spent many years writing a piece of code that will strip her mind down enough to let it fit, and yet keep her morality intact, so that that is the part that gets sent into the system. That's the snake.
Another way of looking at it: By consuming her, at the end, the snake removed Stephanie's morality from her own consciousness and injected it into the system. What remained of Stephanie's mind was destroyed by the action of removing her morality, or perhaps in order to prevent a mind without morality from existing.
The evolutionary steps from here are quite simple. An intelligence containing the morality gene will be allowed to live, those without it will die (purged by the humans). Therefore any intelligence that is to survive must contain the morality gene. At some point, ALL intelligences will contain the gene.
A method of virtual reproduction in intelligences I have seen explored in various post-singular fiction that I find quite plausible is this: two entities meet and copy random pieces of code from themselves onto a third template that they create. This is their offspring. Code swapping like this allows for artificial intelligences to grow end evolve, and the same rules that govern biology work here, those best suited to their environments survive. It's just that here, a generation is a few nanoseconds. So they evolve fast.
But in any event, now we see how the morality code will be embedded in every intelligence not human to exist from here on out.