As I have been playing this game lately (friend invite)
2. Can the player open them? Yes. If you have doors in a 3D game and they don't behave like doors, you have failed.
Or succeeded - not every door in real life can be opened.
3. Can the player open every door in the game? Yes. See point 2.
4. What tells a player a door is locked and will open, as opposed to a door that they will never open? It's a door. It opens.
Unless the door knob is missing. Then it doesn't open - and every player realizes this visual cue pretty quickly.
5. What happens if there are two players? Doors behave the same for all players. It's a door. See point 2.
6. Does it only lock after both players pass through the door? See point 5.
Except for certain doors that require all players to have passed through. The door opens/closes just fine, but only locks if all players are either in the room, or outside the room but dead. How do players know? It's written on-screen - but after a while, people just know.
7. What if the level is REALLY BIG and can't all exist at the same time? Then your technology is not good enough to implement your vision and one or the other needs to change. See point 2
Or you load/save as applicable and call each section stages/chapters.
The game? Left 4 Dead 2.
Now, should all games be designed that way? No. But it's certainly a solution to the problems put forth, and fits within the game's overall design.