Does taking a screen capture and screen recording on the iPhone X include the "extra 2 small regions (notches)" at the top? And can it be turned off to only include the main screen area?

(I think the small region is called a "notch"):
The screenshot will include whatever is supposed to be there.

For example if you're on your home screen it will show the part of your wallpaper you cannot physically see when looking at the device.

If you're watching a video horizontally on YouTube for example, it will show whatever is supposed to be displayed in the notch too.

While watching videos horizontally you actually lose out on what is supposed to be displayed in the notch, rather than scaling the video size down to accommodate what would be a smaller res video.
If something vital was displayed in the non-visible notch part of the video, you could take a screenshot of the frame, view the part you needed, then delete the screenshot without saving it for example.

You can see an example of a screenshot on an iPhone X here.