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/lib/router.svelte.ts

@@ -7,6 +7,8 @@ export type RouteName =
   | 'lobby'
   | 'new'
   | 'game'
+  | 'gameChat'
+  | 'gameCheck'
   | 'profile'
   | 'settings'
   | 'about'
@@ -29,7 +31,10 @@ function parse(hash: string): Route {
     case 'new':
       return { name: 'new', params: {} };
     case 'game':
-      return seg[1] ? { name: 'game', params: { id: seg[1] } } : { name: 'notfound', params: {} };
+      if (!seg[1]) return { name: 'notfound', params: {} };
+      if (seg[2] === 'chat') return { name: 'gameChat', params: { id: seg[1] } };
+      if (seg[2] === 'check') return { name: 'gameCheck', params: { id: seg[1] } };
+      return { name: 'game', params: { id: seg[1] } };
     case 'profile':
       return { name: 'profile', params: {} };
     case 'settings':