feat(messages): standalone Messages module + floating chat dock

Move messaging out of the notification center into its own left-nav module
(own page, own unread badge), so a reload returns to it and notifications stay
separate. Add a Messenger-style floating dock (bottom-right): pop a conversation
out from the page, keep chatting across navigation, minimise (rollup) or close.

- Messages is now a routed page; NotificationsPanel reverts to inbox-only and the
  nav badge no longer mixes in messages.
- Extract shared KeyGate + ChatThread (used by both the page and dock windows);
  e2ee exposes a shared unlock subscription so page and dock agree.
- ChatDock (always mounted for human users) owns the app-wide live-message
  subscription and per-device key restore. EN/HU/DE strings.

frontend/src/components/NavSidebar.tsx

@@ -12,6 +12,7 @@ import {
   Info,
   ListVideo,
   LogOut,
+  MessageSquare,
   Settings,
   Shield,
   SlidersHorizontal,
@@ -126,15 +127,14 @@ export default function NavSidebar({
   // One indicator for both layers: durable server notifications + the client-side transient
   // events (the former separate bell is now folded into the inbox page).
   const clientUnread = useSyncExternalStore(subscribe, getUnreadCount, getUnreadCount);
-  // Unread direct messages also live in the notification module, so they add to the same badge.
-  // The demo account can't use messaging, so don't poll it there.
+  const unread = (unreadQuery.data?.count ?? 0) + clientUnread;
+  // Direct messages are their own module with their own unread badge (demo can't message).
   const msgUnreadQuery = useLiveQuery(
     ["message-unread"],
     api.messageUnreadCount,
     { intervalMs: 30000, enabled: !me.is_demo }
   );
-  const unread =
-    (unreadQuery.data?.count ?? 0) + clientUnread + (msgUnreadQuery.data?.count ?? 0);
+  const msgUnread = msgUnreadQuery.data?.count ?? 0;
 
   type NavItem = { page: Page; icon: typeof Home; label: string; badge?: number };
   // User-facing content modules vs. admin/system modules, separated by a divider in the rail.
@@ -143,6 +143,10 @@ export default function NavSidebar({
     { page: "channels", icon: Tv, label: t("header.account.channels") },
     { page: "playlists", icon: ListVideo, label: t("header.account.playlists") },
     { page: "notifications", icon: Bell, label: t("inbox.navLabel"), badge: unread },
+    // Direct messaging — its own module; hidden for the shared demo account.
+    ...(me.is_demo
+      ? []
+      : [{ page: "messages" as Page, icon: MessageSquare, label: t("messages.navLabel"), badge: msgUnread }]),
     // Per-user sync status + your own API usage (admins get an extra system-wide tab inside).
     { page: "stats", icon: BarChart3, label: t("header.account.stats") },
   ];