feat(api): add auth middleware to protect game endpoints

- Add requireAuth middleware using Better Auth session validation
- Apply to all game routes (start, answer)
- Unauthenticated requests return 401

apps/web/src/lib/auth-client.ts

@@ -0,0 +1,7 @@
+import { createAuthClient } from "better-auth/react";
+
+export const authClient = createAuthClient({
+  baseURL: "http://localhost:3000",
+});
+
+export const { signIn, signOut, useSession } = authClient;