1. Are you flying AA in F or J? If so, avoid the Flight Connections Center and just clear Immigration and Customs (if in F or J, you can use the Fast Track immigration line) in Terminal 3 and then head over to the correct terminal for your connecting flight (Terminal 1 or 4, depending on your destination). Since you are traveling in J on BA, you can use Fast Track for clearing security in the correct terminal. You still have to clear security, but you cna avoid the line. If you are leaving from Terminal 1, you can walk there from Terminal 3 using LHR's underground corridors. It is well signed and easy to find, but a bit of a walk. If you are going out of Terminal 4, you can get there landside using the Heathrow Express, which is free between the Terminals. If you have the time, and are in F or J on AA, you cna use AA's Arrivals lounge in Terminal 3 for a shower or BA has showers in its Terraces lounges in Terminal 1 and 4.

2. Definately call BA back and try again for a seat assignment. They most definately do assign seats in Club Europe. As JohnAx said, go for the A and C seats as there are only two on that side and they are a bit wider than the other seats and do not have a shared arm rest. They are not all that great, but better than the D,E and F seats.