Not just in snow, but in general, in cold, battery runs out much faster.
If it is not snowing, you are okay. However, your kit lens is not water sealed, so you must protect it from snow similarly to protecting from rain.
I do not think that would be necessary. However, check the specs of your battery. (Minimum operational temperature, minimum storage temperature.) A camera turned off still counts as operational, however, very little current is being drawn. People take out the batteries to put it in a pocket so it gets body warm, so that it could source current again, not to protect them from damage I guess.
Your camera and lens is not water sealed. If it is snowing, use the same protective means you would use in rain.
And one, most important advice from me: your biggest enemy is CONDENSATION. That is, humidity inside your camera becoming water drops when you suddenly move your camera from a hot area to a cold one or the other way, from cold to hot. You should gradually cool down or warm up your camera, or allow time for that condensed water to evaporate. There are instructions in your User Manual how long this takes.