Phase 1: Database and API updates
- Create migration 0010 to make game_name and game_config nullable
- Update arcade_rooms schema to support rooms without games
- Update RoomData interface to make gameName optional
- Update CreateRoomParams to make gameName optional
- Update room creation API to allow null gameName
- Update all room data parsing to handle null gameName
This allows rooms to be created without a game selected, enabling
users to choose a game inside the room itself. The URL remains
/arcade/room regardless of selection, setup, or gameplay state.
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Co-Authored-By: Claude <noreply@anthropic.com>