RCS8551 - Can you expand upon the "infamous CPH fare" thing.

I fly CPH at least once/twice a year and am curious what this fare is, what class of service, how much etc. blah blah blah.

While I typically fly JFK-CDG-CPH, I would love to fly JFK-AMS-CPH just to avoid CDG. Can I do this also on award travel and/or fly to BLL instead of CPH? Many thanks in advance as we are beginning at looking for flights next summer.