Refactor settings to use FirebaseAuth for sign out

Replaces the previous mock-based authentication in client settings with FirebaseAuth for sign out functionality. Updates dependencies and imports accordingly, and adds firebase_auth to the settings package and firebase_core to the client app. Also includes minor Dart type improvements and dependency reordering.

apps/mobile/apps/client/pubspec.yaml

@@ -10,11 +10,6 @@ environment:
 dependencies:
   flutter:
     sdk: flutter
-  cupertino_icons: ^1.0.8
-  flutter_modular: ^6.3.2
-  flutter_bloc: ^8.1.3
-  flutter_localizations:
-    sdk: flutter
 
   # Architecture Packages
   design_system:
@@ -31,6 +26,13 @@ dependencies:
     path: ../../packages/features/client/settings
   client_hubs:
     path: ../../packages/features/client/hubs
+  
+  cupertino_icons: ^1.0.8
+  flutter_modular: ^6.3.2
+  flutter_bloc: ^8.1.3
+  flutter_localizations:
+    sdk: flutter
+  firebase_core: ^4.4.0
 
 dev_dependencies:
   flutter_test: