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/packages/features/client/settings/lib/src/data/repositories_impl/settings_repository_impl.dart

@@ -1,19 +1,23 @@
-import 'package:krow_data_connect/krow_data_connect.dart';
+import 'package:firebase_auth/firebase_auth.dart';
 import '../../domain/repositories/settings_repository_interface.dart';
 
 /// Implementation of [SettingsRepositoryInterface].
 ///
-/// This implementation delegates data access to the [AuthRepositoryMock]
-/// from the `data_connect` package.
+/// This implementation delegates authentication operations to [FirebaseAuth].
 class SettingsRepositoryImpl implements SettingsRepositoryInterface {
-  /// The auth mock from data connect.
-  final AuthRepositoryMock mock;
+  /// The Firebase Auth instance.
+  final FirebaseAuth _firebaseAuth;
 
-  /// Creates a [SettingsRepositoryImpl] with the required [mock].
-  SettingsRepositoryImpl({required this.mock});
+  /// Creates a [SettingsRepositoryImpl] with the required [_firebaseAuth].
+  SettingsRepositoryImpl({required FirebaseAuth firebaseAuth})
+    : _firebaseAuth = firebaseAuth;
 
   @override
-  Future<void> signOut() {
-    return mock.signOut();
+  Future<void> signOut() async {
+    try {
+      await _firebaseAuth.signOut();
+    } catch (e) {
+      throw Exception('Error signing out: ${e.toString()}');
+    }
   }
 }