I've stripped the coating on my CS knives using 3 different methods. Steel wool, chemical stripper, and just scrapping it off with another hard object (in my case another knife blade). All three got the job done, but I strongly prefer the steel wool method personally. The stripper is fine, but it is messy and has some cost associated with it (ie, you have to buy the stripper and best to buy some chemical resistant gloves to be safe). Scraping it off with another hard object (like a knife or bottle opener) works too... but I liked the aesthetic results I got with the steel wool better (ie, it left the stonewashed blade underneath with a cleaner, softer sheen).
The steel wool method does take a little more time though... but worth it in my book. YMMV.
Oh yeah... and definitely take the knife apart... and don't worry about leaving any coating around the pivot area... removing it doesn't adversely affect the pivot in any way. Also, I did the lockbar and pocket clip too with steel wool... in the case of the pocket clip the steel wool method will leave the coating in the "Cold Steel" logo on the clip... which is a nice look IMO.