Which sensors to choose to detect movement direction
The PIR sensors can work in your case. They are passive and detect only moving objects that emit IR radiation. So anything that's hotter than background like people, cats, dogs but not doors.

The raw sensor looks like this:



But it's usually available as a part of an IC that includes a plastic lens that focuses IR rays from broader area on the sensor.



Remove that lens (or not) and add something made of cardboard to limit the area that sensor sees and you have a directional sensor that detects moving hot things.

Technically you can completely hack the raw sensor and get info if the hot object is moving left or right. Read this article and look at the sensor response graph.