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/src/game/Chat.svelte

@@ -69,10 +69,13 @@
     display: flex;
     flex-direction: column;
     gap: 10px;
-    /* dvh so the chat shrinks with an open keyboard, keeping the start of the
-       conversation on screen instead of pushed above the fold (vh fallback). */
-    height: 56vh;
-    height: 56dvh;
+    /* Fill the chat screen; the list scrolls and the input pins to the bottom. The screen
+       fits the visual viewport (--vvh), so an open keyboard simply shrinks it and the input
+       stays visible — no modal relayout, no page jump (Stage 17). */
+    flex: 1;
+    min-height: 0;
+    padding: 10px var(--pad);
+    box-sizing: border-box;
   }
   .list {
     flex: 1;