Play a friend
One link for every game. For Strategy and two-player Classics you take turns, whenever suits you, with no clock. For puzzles you play the same board and compare scores. No chat, no messages: the board is the conversation.
Six screens, on a phone. Signed-in desktop gets the same thing in the sidebar: a Your turn entry above Today.
Three calls for you. 1. Can a friend take their first turn without an account? I'd say yes, then ask them to sign in to keep the game. 2. When it's your turn and you're not on the site: nothing (Home shows it), an email, or a notification. I'd start with nothing and add notifications with the app. 3. A game nobody moves in for 7 days ends as abandoned. Right length?