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/ws/connections.ts

@@ -0,0 +1,44 @@
+import type { WebSocket } from "ws";
+
+// Map<lobbyId, Map<userId, WebSocket>>
+const connections = new Map<string, Map<string, WebSocket>>();
+
+export const addConnection = (
+  lobbyId: string,
+  userId: string,
+  ws: WebSocket,
+): void => {
+  if (!connections.has(lobbyId)) {
+    connections.set(lobbyId, new Map());
+  }
+  connections.get(lobbyId)!.set(userId, ws);
+};
+
+export const removeConnection = (lobbyId: string, userId: string): void => {
+  const lobby = connections.get(lobbyId);
+  if (!lobby) return;
+  lobby.delete(userId);
+  if (lobby.size === 0) {
+    connections.delete(lobbyId);
+  }
+};
+
+export const getConnections = (lobbyId: string): Map<string, WebSocket> => {
+  return connections.get(lobbyId) ?? new Map();
+};
+
+export const broadcastToLobby = (
+  lobbyId: string,
+  message: unknown,
+  excludeUserId?: string,
+): void => {
+  const lobby = connections.get(lobbyId);
+  if (!lobby) return;
+  const payload = JSON.stringify(message);
+  for (const [userId, ws] of lobby) {
+    if (excludeUserId && userId === excludeUserId) continue;
+    if (ws.readyState === ws.OPEN) {
+      ws.send(payload);
+    }
+  }
+};