1. Depends totally on the route and the given flight. Status pax can preassign. Also, many people will have been preassigned a seat. You will get choice, though expect any status passengers to have grabbed any popular seats.

2. If you are through ticketed then, assuming your onward flight is a connection within 24 hours (not a stopover), you can checking for the onward flight(s) at the same time as the first one - i.e. T-24 of the first departure time.
Therefore, on your second legs you will have a good few hours jump on non-status passengers who are beginning their journey at LHR!
Remember though that some people will be preassigned - families etc. And that not all the plane is available to choose form.

OBVIOUSLY IT IS 24 HOURS BEFORE LOCAL DEPARTURE TIME - otherwise everyone would claim they lived in Australia. [Unless a connection as I said above]

3. No chance. You elected to travel when you did. There is no requirement to checkin at exactly 24 hours before flight and, regardless, much like the departure time, the time online checkin opens is a property of the given flight you booked.
N.b. British terminology, this is a 'Coach' (also this, don't get mixed up). You are in WT (Economy) or WTP (Premium Economy).

4. London? It's busy all the time. The behavior of this specific route, I've no expert knowledge on.

5. This is AA question. As far as BA are concern no, it's not their currency. I've no idea about AA scheme as I'm not part of it. However, given that it sounds like you are ticketed already, it would certainly depend upon fare rules. Purchasing channel might well come into it. And, generally speaking, I'd very much doubt it. But it is an AA question, not BA.