Wirelessly posted (BlackBerry8530/5.0.0.601 Profile/MIDP-2.1 Configuration/CLDC-1.1 VendorID/417)

You can redeem BA miles for one-way awards, so there is no problem flying into one airport and out of another. If you start your return trip from some country other than the UK, you will avoid the high UK air-passenger duty on departing flights, as long as you do not stop over in London on your return trip. (Conections via London are OK.)

You would have to pay "taxes" even if you redeemed AA miles for AA flights. It is BA's fuel surcharge that you will have to pay when you fly AA or BA TATL.

You are entitled to free en-route stopovers when redeeming BA miles, even on partner carriers. So if you would like to break up your trip in BOS, NYC, WAS, etc., you should be able to do so, assuming you can find award-seat availability. (On your return, you could make LAX a stopover, and then get a free one-way to Hawaii on AA, on the same award ticket.)

And unless you are also getting an award ticket for your baby, I think you will have to pay a significant sum for a ticket for the baby. Babies typically do not fly for free on international flights. Check with BA for more info about this.