Should I caulk around this drain pipe?
You should seal that hole to keep vermin from passing through and to insulate. Any traps (the "u" or "p"-shaped drain fixtures that maintain a water-seal to prevent sewer gases from venting into your home) should be prevented from freezing, but those are usually located under the sink fixture outside of the wall. Some bathtubs and showers may be an exception to this, but in freezing locales these traps are usually placed in a location normally protected from freezing (like an interior wall or basement).