Regarding the use of the pedal throughout this piece, you can get away with a syncopated pedal (that pedal only to connect notes legato but not to sustain notes), do not let notes overlap one another with the pedal but with the held down fingers instead.
My student has been working on this piece for over half a year and could easily play the piece in a week using the pedal, the great difficulty came when playing it with a legato touch only with the fingers, this I think is the challenge we are faced with and it is only now I can start hearing it being played right when the pedal is removed. With excessive legato from the pedal you literally ruin the sound Shostakovich is trying to present (even though it does sound nice too it isn't what was intended.) The simplicity of sound and the sustaining of notes with the finger is what makes this piece sound absolutely lovely in my opinion.
I never touched this piece prior to teaching it, and I for one found this peice extremely interesting when we consider our fingering. Fingering becomes paramount and extremely difficult when we want to do everything without the pedals help. This really shifts us into a different state, and a very very difficult challenge.
Like Bach WTC, we do not need to bring out this, get louder here or this or that. The notes stand alone and a natural tendancy (to get louder as notes move up the keyboard for example). We make our playing sound fake if we start putting volumes here and there. The same applies for the Shostakovich P+F since I think he is trying to echo Bach in this book.
Some parts just need the pedal, especially if you have small hands. Reaching the the large 10ths later on in the peice can be impossible for most. My student has small hands so he must use the pedal to connect the sound with the large stretches, luckily they don't last long.
What about the tempo of the peice? Why do I always feel like it should be taken slower than how it is usually played? When I play this I play it quite slowly, probably 70% of the recording of this post. I guess everyone has their own ideas.