Yes, TXL will be kept operational for another six months. Don't know about actual flights, though. Last day of regular ops will be Nov 7. On Nov 8, a final Air France flight will depart to Paris CDG. First scheduled flight to land at TXL in 1960 also was an AF.
Tegel was in the French Sector of Berlin, so of course AF was the first and now appropriately the last. Tempelhof was in the American Sector, and Gatow in the British Sector. In the Soviet Sector was Adlershof. Contrary to much belief, Schonefeld was not in the Soviet Sector, or in East Berlin, but (just) outside, in the mainstream DDR, which led to all sorts of additional bureaucratic issues to get there from West Berlin.