If you're sure there is no leak, then the problem is not just insulating the pipes, but the small room. Or at least get some heat/air circulation in there. Sounds like the humid air from the house is condensing on the entire small room creating the moisture/ice on the walls and sweating pipes.
You need to dry out the room and then insulate it. It would help to have some air circulation from the basement in there as well.
Fixing the warping of the wood floor will be another big project if it is warped badly.