As to the ticketing carrier, my understanding has always been that it's the marketing carrier of the first leg ....with some exceptions, one being that AA issues tickets on behalf of JL (or at least they used to and, by the sound of it, still do). If the first leg is on BA, then BA would issue the ticket though I seem to recall there being some issue with having to call them to give them the card number.
There is no rule that requires that.