Showers with half a door... why?

Aren’t those usually wet rooms (i.e. drain in the bathroom floor instead of only inside the shower)? Often with the drain along the shower wall and the floor with a slight slope that makes most of the water go that way naturally.

Yeah, the ‘wet room’ concept is growing in popularity; typically, you ‘tank’ the whole room to make it watertight, then tile the whole thing, then it doesn’t matter if you trail water out of the shower or bath, because the whole room is the cubicle.
I stayed in a holiday cottage that had just such a wet room setup, it felt weird to be showering in what was essentially just a corner of the room, but it seemed to work OK.

Some yes, some no. I’ve seen it both ways.

For sure in a room w no lip between the shower and the main area they expect some water outside the shower. Whether they expect it all the way to the far wall, and whether you could clean the entire room by just hosing it out like you might a driveway is often a different matter.

I stayed in a hotel overseas with a wet room bathroom and the configuration made it difficult to ensure that the toilet paper stayed dry.

I don’t really like the wet room concept. I like the floor to be dry where i stand to brush my teeth. But also, if it’s a wet room, why even half a wall between the shower and the rest of the room? The wet room bathrooms I’ve been in had no hard separation of the shower area. Maybe a curtain.