Flights to GVA are out of T5, so taking the BA route (PHX-LHR) will mean you stay in the same terminal, whereas the AA route (PHX-LAX-LHR) will get you into terminal 3, requiring a bus transfer to T5. (You want to think JFK rather than DFW in terms of how the terminals are (not) connected.)

If on one ticket, baggage will be checked through either way. Next time you see your luggage is in Geneva.

In terms of service, in Y I would say there is not much difference between AA and BA, especially since you'll want to sleep most of the way.