1. Camera temperature
First of all, most cameras are rated to about 32 degrees Fahrenheit by the manufacturer, so if you are going into conditions that are around this or higher than this, you really don't need to pay attention to the cold as much. Of course, you do still need to worry about precipitation. I personally have taken gear in colder weather and have been fine, but 32 degrees is what most manufacturers rate the gear to.
In the cold weather, batteries are usually one of the first things to go. These magical devices are just not meant to be used very effectively in low temperatures, so make sure you have a few on you. I would also recommend that you keep your spare battery somewhere that is warm, such as a a place near your body like an inner coat pocket. If you are camping in the cold, then this goes for the camera as well. I usually keep mine in the tent and in a bag under some clothes to maintain a warmer temperature.
Of course no camera likes being in the snow or rain. Many of the higher-end camera bodies and lenses are water sealed so they can get a few drops on them without any problem. That being said, you want to protect your investment and stop any damage the elements can surely cause. You can consider low-cost bags that will protect the camera by wrapping it in plastic and providing a string to tighten it around the lens. You can also use more expensive covers that actually conform to the camera's body and provide a waterproof fabric that protects the camera. Of course, I would be remiss if I didn't also state that a simple plastic bag can provide some protection in the rain and snow, but be weary of leaving your camera under something like this for a long period of time.
One of the things that most non cold weather people forget is just how often their hands will be outside of their coat when shooting. I have forgotten gloves on a few sunrise shoots and can tell you it was miserable. That being said, gloves can be a little annoying when trying to control your gear, so I would recommend investing in a pair with touchscreen finger tips or buying a pair of free-finger gloves. The latter of which is not great if you are in severe cold, as you do not want your fingers exposed for long periods of time. If you have a favorite pair of gloves but they aren't touchscreen compatible, take a look at this video on how to make your own gloves work with a touchscreen device.
Another consideration I would recommend is a good UV filter. These filters don't do much to change the appearance of the photo itself (other then to remove some haze), but they do make it so your lens is much more protected from the elements. I usually have a UV filter on my camera almost all of the time to keep it protected, but in the cold weather it is a must.
This one really should go without saying but you need a good camera bag when you are shooting in the cold weather. I like to use a sling type bag that I can easily put under my jacket if I need to, which allows me to keep my gear covered and warm when I am not using it. I leave my camera in this all of the time when I am outside if I'm not shooting, as it helps with keeping it out of the cold and condensation.
That completes my main considerations for when shooting in cold weather. I would love to hear what you think I left off in the comments!
*All photos in the body of this post were taken by Josh McNair. If you like his style, follow him via @californiathroughmylens on Instagram!