If you use back button focusing, it is like combining AF-S and AF-C together. Unless you want the camera to decide on what to focus on, do not use AF-A.

This is a starting point that Scott Kelby uses:
Amount=50
Radius=1.4
Detail=15
Masking=60

From here you can make your own tweaks.

I try to avoid iso 25600 but I have shot at iso 20000. If I got the D4s, I would have no problems shooting at 25600. If you ever shoot high school football in the evenings you would understand. This settings on this shot were 1/800, f/2.8, iso 20000. The exposure was boosted a touch more in post processing also.

ISO 20000 shot

I would attack this the same way I would shoot any athlete. My shutter speed would be at least 1/800 to 1/1000 to stop all motion, I would have my aperture no wider than f/5.6 but preferably around f/2.8 to isolate my subject. I would then adjust ISO for exposure. Make sure you are using af-c single point auto-focus so you are capturing the dog and not the poles. Try to shoot from a much lower angle to make the dog look more prominent and dynamic. If you want to have a cool panning shot, sho0t at 1/25 of a second or lower and try to follow. Your hit rate will be very low, but the ones you do capture will look amazing.

When using flash, you are actually adjusting two exposures. The exposure for your subject and the exposure for your background. The subject is controlled more by your aperture and your background is controlled by the shutter speed. I would set camera to manual, iso 200, f/5.6, ss 1/125 flash at 1/2 power. Make sure to get your flash close to your subject. The closer to your subject the flash is, the darker you can get your background do to the inverse square law. Now, take a shot. If your subject is too bright or too dark, adjust the aperture. To darken your background increase your shutter speed.

I think I would have narrowed your depth of field in this shot to separate it from the background. To make it pop, some work in post is necessary.

This is my attempt

Since most of my shots are done with longer glass and/ or with wide open apertures of f/1.4 to f/2.8, fine tune adjustments are almost a necessity because the depth of field is so shallow. If you are shooting at f/4 and above, it might not be as important.

georgevedwards If you manually zoom in and focus, then zoom out, your image will be out of focus. What I have done is compose my image, then go to live view and zoom in with live view to focus manually, then get out of live view and snap the picture. Works great.

If you are serious about wanting sharp photos, you will be glad that your camera has the ability to focus fine tune. The mount that you attach your lens to can be off by microns whether it is caused during the manufacturing process or the repeated changing of lenses by the photographer, but this is the reason for the ability to fine tune. Zoom lenses don't benefit from this as much as primes do. With zooms, adjust at the focal length that you most commonly use. I have used the "Dot Tune" method and found it to be very easy and accurate. Here is the link if anybody is interested. http://www.fredmiranda.com/forum/topic/1187247

Shooting in the gyms most kids play in is a challenge since the lighting is not the best. Since this is the case, a lens with an aperture of at least f/2.8 is usually required to get a shutter speed of 1/500 or greater to minimize motion blur. If you are able to get to the baseline and have the action come towards you, you can get away with a shutter speed of 1/320 but you will still have some blur in the moving appendages. My typical setting for a high school game is f/2.8, 1/640-1/1000, and an iso of 6400. I know you said you didn't want to buy new glass, but a 50mm lens with an aperture of f/1.8 can be had for around $100 and would be perfect for the gyms. I can get settings of f/2, 1/1000, iso 2500 with this set up. Set AF-C and single point to make sure you are focused on your subject and aim at the chest. One more thing, if you time the shot at the peak of the jump, you can also minimize motion blur in your subject.

To get sharp prints you must first have a sharp image. Since you are using continuous lighting which will not freeze action, you have to have a higher shutter speed to reduce movement. Using the one to one ratio of focal length to shutter speed (50mm to 1/50 sec)or using image stabilization will help reduce camera shake. To stop movement of the pets, 1/250 or higher would be required.