As for the heat, I'm gonna have a duct go directly to the closet, so hopefully no problems there.

Duct to the central heating/AC or duct to the outside of the house? I am not sure a standard duct to your furnace/AC would work. There is quite a bit of heat produced, and like Tony said, you dont want to be adding heat when the furnace is on. A couple bathroom-style exhaust fans are cheap and easy to install.