Nope, depends on the software, there are other solutions available.
Not allways, you can make it non-auth. But if you want to prevent people cheating it’s the way to go.
All provide some sort of ‘room’ architecture to group players in a game.
Not realy, 1 server handles all the games. Usualy if your using middleware like SFS or Photon the server won’t have a player.
The client/server coding is sepperated from eachother and the server doesn’t have to have a player.
In your case the server would be authoritative and decide which player’s turn it is and inform both players about it. When a players makes a move the server will make sure his turn is over and enable the other player to play and so on. I was talking about networking in general where you don’t need everything validated by the server. You can have for example a player that shoots a bullet which would be validated by the server so he can’t cheat but you can also have a player do other things in the game that doesn’t need server validation.