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/src/server.ts

@@ -1,8 +1,13 @@
+import { createServer } from "http";
 import { createApp } from "./app.js";
+import { setupWebSocket } from "./ws/index.js";
 
 const PORT = Number(process.env["PORT"] ?? 3000);
 
 const app = createApp();
+const server = createServer(app);
+
+setupWebSocket(server);
 
 app.listen(PORT, () => {
   console.log(`Server listening on port ${PORT}`);