Chat + word-check as their own screens; in-game unread badge (review item 7)

- Chat and word-check are now routed screens (/game/:id/chat, /game/:id/check) with a
  header back to the game and no tab-bar, replacing their modals. The soft keyboard just
  resizes the visible viewport (tracked into --vvh, which the Screen height uses since iOS
  does not shrink dvh for the keyboard) with the input pinned to the bottom: no modal
  relayout, no page jump. Supersedes the earlier bottom-sheet Modal attempt.
- A new chat message raises an unread badge on the in-game hamburger + the Chat menu row
  (per game, cleared on opening the chat), mirroring the lobby badge.
- TG native back + the header back chevron return chat/check to their game.
- Exposes --tg-safe-top (device notch) for the finalised TG-fullscreen header.

Tests: e2e for chat/check opening as their own screens + back. Docs: PLAN, FUNCTIONAL(+ru).

ui/e2e/game.spec.ts

@@ -162,6 +162,25 @@ test('a placed tile drags from one board cell to another (Stage 17 relocation)',
   expect(to).not.toBe(from);
 });
 
+test('chat and word-check open as their own screens and back to the game (Stage 17)', async ({ page }) => {
+  await openGame(page);
+
+  await page.locator('.burger').click();
+  await page.getByRole('button', { name: /^Chat$/ }).click();
+  await expect(page).toHaveURL(/\/game\/g1\/chat$/);
+  await expect(page.locator('.pane')).toHaveCount(1); // let the slide transition settle
+  await expect(page.locator('.chat')).toBeVisible();
+  await page.locator('.back').click(); // header back chevron returns to the game
+  await expect(page).toHaveURL(/\/game\/g1$/);
+  await expect(page.locator('.pane')).toHaveCount(1);
+
+  await page.locator('.burger').click();
+  await page.getByRole('button', { name: /Check word/ }).click();
+  await expect(page).toHaveURL(/\/game\/g1\/check$/);
+  await expect(page.locator('.pane')).toHaveCount(1);
+  await expect(page.locator('.check input')).toBeVisible();
+});
+
 test('the board-label mode in Settings changes the on-board labels', async ({ page }) => {
   await openGame(page);
   // beginner (default) renders split "3× / word" labels.