With regards to a Jewish man wearing a yarmulke, is it proper to wear a hat over the yarmulke on a cold day?

It's done all the time. The idea is merely to have your head covered.

This site confirms that wearing a kippah under a hat is not a problem:

https://en.wikipedia.org/wiki/Yarmulke

As far as I know, being Jewish, there is no problem wearing a hat over your Yarmulke or Kippah. Most Chasidic men wear fedorahs and tall hats all the time, and underneath is the kippah, so they remove their hats when they go to schul and they don't need to look for the kippah. Orthodox Jewish men also wear kippah all the time, and they are able to wear hats over their kippah. Other wear hats over their kippah to conceal the fact that they are Jewish. As you move from between the extremes downard in Judaism, Chasidic and Orthodox wear Kippah all the time, Conservative wear kippah during prayer and services, Reformed wear kippah optionally during prayer. These are generalizations, and there there are many grey areas and nuances in between.