To answer 2 of my questions:

1: Yes, it is a BE fare. Seat selection is possible, but requires payment on the BA flights (BA HBO Fare). Why Google Flights doesn't have a basic economy filter is beyond me.
3: This is a UI glitch (due to the new interface?) and is actually a BE flight. The Fare Bases are actually BA E (Basic) fares (according to EF)

I was able to get the original ITA Matrix fare to be bookable on AA.com by having AA marketed and AA operated domestic flights, AA marketed and BA operated transatlantic flights and swapping the Europe flights to BA marketed and BA operated. As to why this worked, no clue, but I did have to force everything by manually putting in the AA and BA flight numbers for each segment.