feat(api): add WebSocket foundation and multiplayer game store

- Add ws/ directory: server setup, auth, router, connections map
- WebSocket auth rejects upgrade with 401 if no Better Auth session
- Router parses WsClientMessageSchema, dispatches to handlers,
  two-layer error handling (AppError -> WsErrorSchema, unknown -> 500)
- connections.ts: in-memory Map<lobbyId, Map<userId, WebSocket>>
  with addConnection, removeConnection, broadcastToLobby
- LobbyGameStore interface + InMemoryLobbyGameStore implementation
  following existing GameSessionStore pattern
- multiplayerGameService: generateMultiplayerQuestions() decoupled
  from single-player flow, hardcoded defaults en->it nouns easy 3 rounds
- handleLobbyJoin and handleLobbyLeave implemented
- WsErrorSchema added to shared schemas
- server.ts switched to createServer + setupWebSocket

apps/api/package.json

@@ -14,12 +14,14 @@
     "@lila/shared": "workspace:*",
     "better-auth": "^1.6.2",
     "cors": "^2.8.6",
-    "express": "^5.2.1"
+    "express": "^5.2.1",
+    "ws": "^8.20.0"
   },
   "devDependencies": {
     "@types/cors": "^2.8.19",
     "@types/express": "^5.0.6",
     "@types/supertest": "^7.2.0",
+    "@types/ws": "^8.18.1",
     "supertest": "^7.2.2",
     "tsx": "^4.21.0"
   }