This should really be posted in the OneWorld forum, not Trip Reports.
You know you get up to 15 segments on an AONE ticket? You seem to have only listed 8 flights..
Consider flying LHR-SFO/LAX on BA, then backtrack to JFK (or Cancun).
I can see the JAL flight from NYC to Tokyo on the OneWorld planner. It may not operate on the day you are checking. But if it does and you can't see it online, you should still be able to book it on the phone.
You can credit to any OW program, so yes you can earn BAEC avios.
Thanks for the advice. I know 8 flights are little, but we are doing the whole trip in only 3 weeks. Never the less, Spending an extra day or so in the US could be fun and we can experience AA first class this way. Never even thought of doing something like that!
Regarding the JAL flight, somehow they only show me JAL codeshare flights for AA. I've looked on the OneWorld website, but can't find an email address for Oneworld, only the individual airlines...