I know I'm late to this conversation but put a brake on the bike. You can stop faster and easier with a brake than you can with back pedaling alone.
The only reason not to have a brake is image.

As for pedal strikes in corners, it is possible with fixed gears, however track bikes (vs. converted road bikes) have higher bottom brackets, shorter crank arms and narrower Q-factors which all greatly increase the lean angle one can achieve in a corner without having pedal strike. So basically track bikes do not generally have a big problem with pedal strike.

Craig