the severe cold will eventually effect the computer in a negative fashion. The first signs of wear you will encounter will be with the fans, cd drives, and other moving parts.
If you turn the computer on and off daily, or multiple times a day you will seriously reduce the lifespan of your CPU. Processors fail due to thermal changes. IF you allow the computer to get to 0, then back up to 100+ when it is on, it will definitely reduce the lifetime of the CPU. If you leave your computer on most of the time though, it should only have a negative effect on the dc motors in your computer.
Hope this helps