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/lobbyGameStore/LobbyGameStore.ts

@@ -0,0 +1,17 @@
+import type { MultiplayerQuestion } from "../services/multiplayerGameService.js";
+
+export type LobbyGameData = {
+  questions: MultiplayerQuestion[];
+  currentIndex: number;
+  // NOTE: Map types are used here for O(1) lookups in-process.
+  // When migrating to Valkey, convert to plain objects for JSON serialization.
+  playerAnswers: Map<string, number | null>; // userId → selectedOptionId, null = timed out
+  scores: Map<string, number>; // userId → running total
+};
+
+export interface LobbyGameStore {
+  create(lobbyId: string, data: LobbyGameData): Promise<void>;
+  get(lobbyId: string): Promise<LobbyGameData | null>;
+  set(lobbyId: string, data: LobbyGameData): Promise<void>;
+  delete(lobbyId: string): Promise<void>;
+}