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

packages/shared/src/schemas/lobby.ts

@@ -112,11 +112,19 @@ export const WsGameFinishedSchema = z.object({
 
 export type WsGameFinished = z.infer<typeof WsGameFinishedSchema>;
 
+export const WsErrorSchema = z.object({
+  type: z.literal("error"),
+  code: z.string(),
+  message: z.string(),
+});
+export type WsError = z.infer<typeof WsErrorSchema>;
+
 export const WsServerMessageSchema = z.discriminatedUnion("type", [
   WsLobbyStateSchema,
   WsGameQuestionSchema,
   WsGameAnswerResultSchema,
   WsGameFinishedSchema,
+  WsErrorSchema,
 ]);
 
 export type WsServerMessage = z.infer<typeof WsServerMessageSchema>;