You will most likely need a new wrear wheel. One that is built for a lockring or else the cog will spin off when you apply force in reverse. As for the pedals, go with what you have now, the fixed gear will be enough of a change. Yes use a brake. Ride for a while and if you don't use it then take it off. I rode for about 3 months before taking mine off although I live in a flat city. If there were hills here I would have a brake.
__________________
I'm not one for fawning over bicycles, but I do believe that our bikes communicate with us, and what this bike is saying is, "You're an idiot." BikeSnobNYC