These are really tough and fascinating questions, both about what capacities are and what persons are. I hope other panelists might add to what little I have to say. What I will say (briefly, and without checking to see what the relevant literature has to say) is that one way to understand capacities is this:
An object X has a capacity C to perform C-relevant functions or behaviors (C-stuff) if and only if X has a structural organization O such that, when X is in the appropriate circumstances, O allows X to do C-stuff.
For example, I have the capacity to multiply any two numbers 1-13 because some structure in my brain allows me to get the right answer when the circumstances arise (e.g., when I'm asked to multiply 8 x 9 and I am awake and paying attention, etc.).
Having a capacity to do C-stuff can be distinguished from having the potential to develop the capacity to do C-stuff. The potential might involve being in a position (both structurally and environmentally) to develop the relevant structural organization to have the capacity. I have the potential to learn Russian, but I do not yet have the capacity to speak Russian.
So, assuming that some complex organization in our brains is what gives us the capacities for self-awareness that I associated with personhood (in my earlier response you cite), we can see that most adult humans have these capacities and are hence persons, while no fetuses have these capacities and hence none are persons. Most fetuses have the potential to become persons (though about a quarter of fetuses end in miscarriages). Unconscious adults actually have the capacities for personhood because they have the structural organization in place; they just aren't in the appropriate circumstances for them to be exercised. Even while I'm asleep or under anesthesia I still possess the capacity (organization O) to do my multiplication tables. I just can't exercise those capacities. A fetus does not have the capacity to do multiplication tables, nor to does it have the capacity to reflect on mental states and consider its past and future, etc.
So, by my definition of persons, fetuses (though they are human beings) are not persons, while unconscious adults are (and dolphins and apes probably are). One can draw various ethical conclusions from this, and I won't do so here. But IF one thinks that being a person means that one deserves more moral consideration than having the potential to become a person, THEN (if one accepts my view) fetuses deserve less moral consideration than adults (and perhaps some animals).
For more on this issue, see here.