I always use AF-C for wildlife. You can get away with AF-S for perched subjects. I have some bad experience with AF-A and never use it.
Your shutter speed of 1/1000 or faster makes sense for moving subjects. I'd probably be shooting at f/6.3 or f/7.1 with your Nikon 70-300 or Sigma 150-500 since they have a variable aperture. With an f/4 lens I'd be shooting at f/4.5-5.6.
I use 51 points rather than 11 points. This is a recent change because the 11 point approach does not use enough cross sensors. The advantage to 11 point is the ability to scroll across the frame quickly.
For perched subjects I try to use single point AF. For a fishing or hunting subject, I try to use single point as well. For more active subjects I move to Dynamic 9 point. I use Dynamic 9 point for birds in flight.
Back button focus makes sense. I try to prefocus on a portion of the subject in the focus plane - the base of the neck or the shoulder instead of the eye in some cases. The DOF can be extremely shallow so this may mean some misses, but it helps. Don't expect the camera to immediately pick up focus on a subject outside the frame. It is often batter the second or third frame. You need to keep your finger or thumb on the AF-On button during this movement to allow AF to "catch up".
Shoot a little looser to make sure you don't clip a wing or foot on a moving bird. This also gives you a little more DOF and room to crop.
Focsu tracking is better at A3 because it keeps the lock on normal. It still tracks the subject - just maintains AF if another bird or a stick enters the frame temporarily.
I want ISO set to give me a slightly faster shutter speed than I need. That's probably 1/1250 rather than 1/1000 or 1/800. If necessary, I use ISO to get there. I try to avoid going to a high ISO needlessly. ISO 1600 is fine, and ISO 3200 in a pinch, but I try to keep it lower. Your settings are fine. You could raise the minimum setting for auto ISO to 1/200 and not have any impact other than a faster shutter speed in great light.