Couldn't this also mean "their mother"?
Not my daughter! (nor my wife)
Ár = our so the answer a máthair could be either her or their mother in this case.
is there a connection between i gcónaí and i mo chónaí? It seems there must be but I can't fathom it.
Yes, if I understand correctly what I'm told, cónaí basically seems to mean residing, in other words abiding, remaining, staying, and so is also used in conveying the idea of "always", or "at all times".