The children playing in the 1920s and the 1930s would create games themselves with what they had, like for example,if they have only a coin and a piece of wood, they would think how to invent a game with the two items.
That depends on when the board game was produced. Any board game that came out during the height of Beatlemania, in mint condition, probably has collector value. More recent board games (Trivial Pursuit, Monopoly, etc.) will probably NOT increase in face value for many years.
A board game doesn't necessarily need a foldable board like Monopoly to be a board game. Games like Carcassonne, Settlers of Catan, and Gold Mine use tiles to build out a dynamic game board, for example.