I can't say much about this part as I don't play zombies anymore. I used to on one of the old CoD's but I honestly can't remember it that well. One thing I can say though is that I have never seen anything other than just running around in zombies so I am a little confused how wall running and double jumping fits in there.
I'm not saying you're wrong, I just have little experience in zombie mode and what I have played was very basic movement. Maybe I just haven't got to a high enough level?