fix(auth): resolve demo login redirect loop, fix backend API bugs, restore clean login UI, and add comprehensive API docs with headers

src/app/signup/page.tsx

@@ -46,6 +46,7 @@ export default function SignupPage() {
           const loginData = await loginRes.json();
           
           if (loginData.success && typeof window !== "undefined") {
+            document.cookie = `auth_token=${loginData.token}; path=/; max-age=2592000; SameSite=Lax`;
             localStorage.setItem("activeUserId", loginData.token);
             const sessionToken = crypto.randomUUID();
             localStorage.setItem("active_session_token", sessionToken);