First, most sports will allow flash. However, not all venues, leagues, or other controlling bodies will allow it. Basketball, volleyball, and hockey (i.e. mostly indoor sports) allow it due to the inherent low light situations. At typical height and distance, it is not that distracting (moms in the bleachers use thier flashes, albeit less powerful ones). Performers/players generally don't notice unless it is right in thier face.
As for the motion freezing. Motion is frozen based on the amount of time light is hitting the sensor. In a low light environment (gym), no measureable amount of light is hitting the sensor at 1/250 of a second at <ISO800, even at f2.8 (this can vary from gym to gym, but a pretty safe bet for HS gyms). Since there is no light, you cannot see any motion. Add in Einstein at 1/2000 t.1. Light is being introduced for only 1/2000 of a second, much like outside on a sunny day, using 1/2000 shutter speed introduces light to your sensor for 1/2000 of a second. Either way, measureable amounts of light are reaching the sensor for only 1/2000 of a second. Inside this 1/2000 happens to fall within the 1/250 the shutter is open.
Imagine walking through a haunted house with strobe lights. The mosnter walking toward you seems to move choppy. This is because the strobe effect is illuminating just a fraction of a second's worth of moving. The moster still makes the same smooth movement anything else does, but you can only see bits here and there. same principle.
Outside or other high ambient light situations, or if exposure settings allow for more ambient light (i.e. you can see an image with no flash used), the flash will illuminate as normal, but your ambient light will continue to illuminate for the full 1/250 of a second, thus limiting your action freezing to what 1/250 can do.
Also note, I stated the t.1 time for the Einstein example. This is important as most flash manufacturers state t.5 times. T.1 times are typically 3 times longer than t.5 times, and the amount of light given from t.5 to t.1 is significant enough to show blur. That is why Einstein is so good at freezing motion, because the IGBT shuts the light off instantly, making the t.1 and t.5 times virtually identical at most power settings (max power will still act much like an Alien Bee or White Lightning).
This link will help you understand t.1 and t.5 times. Please note, this was written before Einstein and Einstein defies these descripiton.
http://www.paulcbuff.com/forums/viewtopic.php?f=5&t=59
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