Can I hide the player list or chat of my private game from strangers?

You can’t hide the player list: there is no such setting. Anyone who opens the game link sees the game page with its lineup, private or not. Private only keeps the game out of Explore and public feeds.

You can close the chat: Settings, “Privacy”, “My events”, “Game chat”, “Only participants”. By default it is “Everyone”, so anyone can write in your games’ chats.