feat(api): add WebSocket handlers and game state management

- handleLobbyJoin: validates DB membership and waiting status,
  registers connection, tags ws.lobbyId, broadcasts lobby:state
- handleLobbyLeave: host leave deletes lobby, non-host leave
  removes player and broadcasts updated state
- handleLobbyStart: validates host + connected players >= 2,
  generates questions, initializes LobbyGameData, broadcasts
  first game:question, starts 15s round timer
- handleGameAnswer: stores answer, resolves round when all
  players answered or timer fires
- resolveRound: evaluates answers, updates scores, broadcasts
  game:answer_result, advances to next question or ends game
- endGame: persists final scores via finishGame transaction,
  determines winnerIds handling ties, broadcasts game:finished
- gameState.ts: shared lobbyGameStore singleton and timers Map
- LobbyGameData extended with code field to avoid mid-game
  DB lookups by ID

apps/api/src/lobbyGameStore/LobbyGameStore.ts

@@ -1,6 +1,7 @@
 import type { MultiplayerQuestion } from "../services/multiplayerGameService.js";
 
 export type LobbyGameData = {
+  code: string;
   questions: MultiplayerQuestion[];
   currentIndex: number;
   // NOTE: Map types are used here for O(1) lookups in-process.