I have a CS '59 P and a CS '60 P. Both are completely silent whether I touch the pup or not. I'm not real knowledgeable, but it sounds like maybe a ground problem. I don't think it's normal.
The pole pieces on mine stick up about 1.5 mm above the plastic cover. As above, I get no sound at all.
I've read the pole pieces can actually be pushed up or down. I've not tried it and I don't think it's a good idea to test the theory. The only time I've heard of one pair of pole pieces being higher was on the original '57 (I think only 1957s) P Bass and for awhile the AVRI '57s. On those, the "A" string pole pieces were higher. I don't know if any CS pups were made to duplicate that, but I've never heard of any.
.