Stage 8: regression tests for the review-round refinements

Lock the polish behaviours so a future edit surfaces as a failing test:
- backend: UpdateProfile now rejects a bad name layout, an away window over 12h, and
  a malformed offset timezone (confirming it wires the Stage 8 validators); a new
  integration test accepts and resolves a "+03:00" offset timezone.
- e2e (mock): the lobby notification badge count, the play-with-friends required
  game type + invitation send, the in-game add-to-friends flipping to a disabled
  "request sent", the profile-edit invalid-name Save guard, and the chat send/nudge
  icon buttons.

ui/e2e/social.spec.ts

@@ -86,3 +86,65 @@ test('finished game draws an inert footer and trims the live-only menu', async (
   await expect(page.getByRole('button', { name: 'Check word' })).toHaveCount(0);
   await expect(page.getByRole('button', { name: 'Drop game' })).toHaveCount(0);
 });
+
+test('lobby hamburger shows the pending notification count', async ({ page }) => {
+  await loginLobby(page);
+  // One incoming friend request (Rick) + one invitation (Kaya) = 2.
+  await expect(page.getByTestId('menu-badge')).toHaveText('2');
+});
+
+test('play with friends: a game type is required to send an invitation', async ({ page }) => {
+  await loginLobby(page);
+  await page.getByRole('button', { name: /New/ }).click(); // lobby tab bar
+  await page.getByRole('button', { name: 'Play with friends' }).click();
+
+  const send = page.getByRole('button', { name: 'Send invitation' });
+  await expect(send).toBeDisabled();
+
+  await page.getByRole('checkbox').first().check(); // pick a friend
+  await expect(send).toBeDisabled(); // still no game type
+
+  await page.locator('.field select').first().selectOption('english');
+  await expect(send).toBeEnabled();
+
+  await send.click(); // the mock creates it and returns to the lobby
+  await expect(page.getByText('Active games')).toBeVisible();
+});
+
+test('game: add-to-friends flips to a disabled "request sent"', async ({ page }) => {
+  await loginLobby(page);
+  await page.getByRole('button', { name: /Ann/ }).click(); // active game vs Ann
+  await page.locator('.burger').first().click();
+  await page.getByRole('button', { name: /Add to friends: Ann/ }).click();
+  // Reopening the menu shows the item as a disabled "request sent".
+  await page.locator('.burger').first().click();
+  const sent = page.getByRole('button', { name: 'Request sent' });
+  await expect(sent).toBeVisible();
+  await expect(sent).toBeDisabled();
+});
+
+test('profile edit disables Save and flags an invalid display name', async ({ page }) => {
+  await loginLobby(page);
+  await page.locator('.burger').first().click();
+  await page.getByRole('button', { name: /Profile/ }).click();
+  await page.getByRole('button', { name: /Edit profile/ }).click();
+
+  const name = page.locator('.edit input').first();
+  const save = page.getByRole('button', { name: /^Save$/ });
+  await name.fill('Bad__Name'); // adjacent specials — invalid
+  await expect(save).toBeDisabled();
+  await expect(name).toHaveClass(/invalid/);
+
+  await name.fill('Good Name');
+  await expect(save).toBeEnabled();
+});
+
+test('chat send and nudge are icon buttons', async ({ page }) => {
+  await loginLobby(page);
+  await page.getByRole('button', { name: /Ann/ }).click();
+  await page.locator('.burger').first().click();
+  await page.getByRole('button', { name: 'Chat' }).click();
+  // Icon-only controls expose their action through the aria-label.
+  await expect(page.getByRole('button', { name: 'Send' })).toBeVisible();
+  await expect(page.getByRole('button', { name: 'Nudge' })).toBeVisible();
+});