The face recognition cameras catch you on the way out, the camera catches you on the way back in.

Not really, in case of a fire alarm all doors unlock anyway.

When I was a consultant I visited many offices where I needed a badge to go to the loo. It's really common in shared office buildings, the toilets tend to be in the elevator hall outside the badge perimeter. Don't think they were monitoring it though.

My building has the toilets in the fire escape stairs which you can push button out to but badge back in from. You would have some difficulty measuring time since you don't know who pushed the button but you could certainly measure frequency.

It depends upon the office. Where I work, we rent space in a building, and the restrooms are shared by multiple tenants on the floor (each floor has a set of bathrooms), so I (and my fellow coworkers) have to badge out and in to use them.