Will the Google Play Game Services multiplayer allow (turn-by-turn) gameplay like DrawSomething?
You are not talking about asynchronous gameplay (which is explained here).
You are talking about turn-based gameplay. But I think you can relinquish the idea of using Google Game Services, because of the missing realtime gameplay (= you don't need to establish a connection between the clients).
I would recommend using a database, which contains everything relating to the game like the gamestate, the players, who did the last turn and things like this.