There's no chance he's sacked after the derby.
There's a negligible chance he's sacked after Palace. A greater chance he's sacked after Burnley and a good chance/verging on almost certain he's sacked after Newcastle.
That's assuming we lose all those games; it'd leave us on 15 points after 20 games, I think the board would have to pull the trigger as we'd have six winnable games after that in a row until City in late Feb, and if at that point we didn't win a sizeable chunk of those games we'd be in desperate trouble as our run-in is hideous, in that you can make argument for us winning 1 of the last 8.
My guess is he survives fairly comfortably. The reason I think we'll improve isn't necessarily because of Benitez, but because the spine of our team will be back and we'll do what we've done before; grind out results due to moments of individual quality from the few decent players we have.
I think we might get a point out of Brentford, maybe a point from Leicester and/or Arsenal, and I think we beat Palace and Burnley and probably Newcastle, meaning we go into the new year on somewhere between 20 and 25 points. I think we'll then rally to around 35 points by late Feb and end the season on around 45 points, good enough for about 13th. It'll then be recognised all round as the season it was - a write off due to FFP and a terrible squad.