Yes, you can redeem AA miles for travel on a BA flight, if award seats are available on the BA flight. There's no need to call AA; aa.com supports award bookings on BA and on a handful of other AAdvantage partners. There will be hefty cash surcharges payable for a TATL award flight on BA when redeeming AA miles.

Yes, AA does fly to several European cities, some via nonstops from JFK, others requiring a connection somewhere. Google "JFK airport wiki" and you'll see a list of all of AA's nonstop destinations from JFK.