So according to my interpretation, Mission Commander can kill itself to rescue a captured card, so with that in mind, do I still resolve her combat icons when that happens? I assume not because she would have left play, but I could see an argument to be made otherwise.

No. Her reaction is when she is focused to strike, which is the step right before you resolve icons. If she dies due to her reaction, the icons remain unresolved because you never got to that step.

Makes sense. Does the same thing apply to a situation where Dark Memories kills the character it is attached to?

Yes.