Refactor E2E tests to use ID selectors and enhance mock auth injection

- Updated E2E selectors for input fields to use stable IDs instead of `name` attributes, improving reliability and alignment with form field guarantees.
- Refined mock auth state injection in Playwright to establish test store state prior to page load.
- Optimized test clarity and consistency by consolidating selector logic and introducing stabilization steps where necessary.
- Removed redundant `AuthInitializer` mocks and refactored related tests to align with the updated `AuthContext` pattern.
- Enhanced readability and maintainability across affected test suites.

frontend/e2e/test-selectors.spec.ts

@@ -0,0 +1,17 @@
+import { test } from '@playwright/test';
+import { setupAuthenticatedMocks } from './helpers/auth';
+
+test('debug selectors', async ({ page }) => {
+  await setupAuthenticatedMocks(page);
+  await page.goto('/settings/profile');
+  await page.waitForTimeout(2000); // Wait for render
+  
+  // Print all input elements
+  const inputs = await page.locator('input').all();
+  for (const input of inputs) {
+    const name = await input.getAttribute('name');
+    const id = await input.getAttribute('id');
+    const type = await input.getAttribute('type');
+    console.log(`Input: id="${id}", name="${name}", type="${type}"`);
+  }
+});