Let me echo the comment about the plastic bag. If you come in out of the cold, condensation can develop within the camera (weather sealing is not the same as absolute seal) and upon going back out, the moisture will turn to ice. This can be avoided by using the bag for 20-30 minutes when changing temperature.
Don't know about the 1DX, but I've used a 50D in -30 (shooting the Aurora). I concur that the battery life can be decreased and one should keep extra batteries in a pocket near your body. The only issue I had was that after about an hour, the camera seemed to "die." I was a bit concerned, but once coming out of the cold, it came back to life and worked flawlessly for the next two years (camera since sold to help finance the 5DDiii).
Also note that the media can potentially be an issue. I shot the Northern Lights with Sandisk Extreme III CF cards because they are rated to work at such a low temperature. I don't know if this mattered but know I was not going to have another opportunity.