Don't confuse "shutter lag" and "shutter speed." "Shutter lag" is the time between when you push the button and when the camera finally takes the picture. "Shutter speed" is how long the shutter is open while the picture is being taken.
Let's say that my son is running and getting ready to jump over an obstacle. With a long shutter lag, it will be very hard to take the picture. I've got to time it so that I press the shutter button well before he starts his jump and hope that the camera takes the picture at the right time. With a short shutter lag, I press the button at virtually the same instant I want the picture to capture him jumping.
With a long shutter speed, my rapidly moving son will be blurry because the picture didn't capture an instant in time; it captured a period of time during which my son was moving. With a fast shutter speed, he'll be blur-free because I captured a very, very brief period of time during which he didn't effectively move.
Something else to consider for "fast" pictures and that is burst rate and buffer size. Burst rate is the rate at which you can take pictures. Some cameras can take one picture every second. Others can take 10 pictures in one second. It's usually given as a number followed by fps (frames per second). The buffer size is how many pictures the camera can take at full speed before it slows down and has to wait for pictures to be written to the memory card. So a camera might be able to shoot 3 pictures every second, but after it has taken 9 pictures it will slow down because the buffer is full.
Red eye is a problem that all cameras have when the flash is very close to the lens. On a small camera, there are no good solutions. It helps if the subject is staring at a bright light so that their pupils are small. That's why some cameras have a red-eye mode that flashes a few times before it takes the picture. Some cameras also detect eyes and try to automatically "photoshop" out the red-eye for you. The only true solution is to move the flash away from the camera, but that's often not practical.