does the Apple watch fall feature work at night when the person is sleeping, gets up for a bathroom run, and falls?

As long as the feature is activated and the person is actually wearing their watch in bed.

ASC - Late 2012 Mac mini, macOS 10.14.6; Watch, series 5 GPS/cellular, watchOS 6.2.8; iPad Air 2, ipadOS 13.6.1; iPhone 8+, iOS 13.6.1; TV 4th Gen, tvOS 13.4.8;  HomePods, audioOS 13.4.8 and Airport Express w/ Airplay 2

The fall detection is running at all times.

Read more: https://support.apple.com/en-us/HT208944