In regard to call signs. Generally speaking, if BA put their flight number on a CX flight it remains a CX flight and uses the CX call sign.
BA are an interesting example. They have a franchise system as well as code share. In the Franchise system an airline effectively loses its identity and becomes BA. Take BMed as an example. They don't operate their own flights - they only operate their flights under BA flight numbers. But they do remain a separate legal entity and, as such, use a Bmed call sign not BA.