Regarding flash:
Hi-speed synch enables you to use your flash at shutter speeds faster than the usual synch speed (I'm guessing that of the 1DMkIII is 1/250 or 1/300 second). The cost is that for each stop faster than the synch speed, you lose the power and reach of your flash by about 1/3.
So let's say the normal synch speed is 1/250. If you shoot at 1/500, you're at about 66% of the flashes normal power. If you're at 1/1000 second, you lose a third of that, so you're at about 44% of original power, etc.
The wider the aperture, and the higher the ISO, the more effective your flash will be, and the greater the reach.
So how it all comes together will depend on the distance to your subject, along with the other settings mentioned. The Better Beamer will focus the flash and throw it further than usual, but still has to work within the constraints of the variables above.
Hope this helps!