If you’re going to run it through duct work, ensure the cable is plenum rated. For Ethernet cable, the jacket will be stamped “CMP”.

A camera platform small enough to fit a standard door frame isn’t going to be able to hold serious optics. The sensor sizes are going to be too small for good night vision, if nothing else.

The temperature issue is probably an issue of economics. Most people don’t need such a wide range, so off the shelf products don’t support the application, as it would incur additional cost.

Both of these are solvable issues however if you’re willing to use a standard security camera mounted over the door instead. There are many cameras that you could pick with large sensors, heating elements, and more.