When I went treking in Nepal it was back in the days of film, so things where pretty straightforward. Though some of the fancier electronic cameras did die in the cold.
But from my skiing experiences, modern cameras shouldn't have too much trouble with the cold at and below base camp. But if there are cold days, it would be worth keeping a couple of batteries inside an inner layer of your clothing and swap them in and out of the camera. The cold will shorten the life of your batteries. But put an apparently dead battery inside a pocket next to your body for an hour and it will magically come back to life.
You are unlikely to find much electricity for a recharge once you leave Namche Bazar, though things may have changed since I was there 10 years ago.