Do I really need INotifyPropertyChanged?

You could try to refer to the example below.
Table Code :

CREATE TABLE [dbo].[Table]  
(  
    [Id] INT NOT NULL PRIMARY KEY ,   
    [Name] NCHAR(10) NULL  
)  

MainWindow.xaml:

<StackPanel>  
        <ListBox  Name="list" Height="300"  Margin="4,4,4,4"  ItemsSource="{Binding Items}"  SelectionMode="Extended">  
            <ListBox.ItemTemplate>  
                <DataTemplate>  
                    <StackPanel Orientation="Horizontal">  
                        <CheckBox  Name="check"  Margin="3"   VerticalAlignment="Center" Click="CheckBox_Click"   IsChecked="{Binding IsChecked, Mode=TwoWay}" />  
                        <TextBox x:Name="tb" BorderThickness="0" Text="{Binding Name}" />  
                    </StackPanel>  
                </DataTemplate>  
            </ListBox.ItemTemplate>  
        </ListBox>  
        <Button  x:Name="btn" Content="Save" Click="btn_Click"/>  
    </StackPanel>  

MainWindow.xaml.cs:

using System.Windows;  
using System.Windows.Controls;  
using System.ComponentModel;  
using System.Collections.ObjectModel;  
using System.Data.SqlClient;  

namespace ListBoxWithCheckBoxTemplateDemo  
{  
  public partial class MainWindow : Window  
  {  
    public MainWindow()  
    {  
      InitializeComponent();  
      DataContext = new ViewModel();  
    }  
    private void CheckBox_Click(object sender, RoutedEventArgs e)  
    {  
      var cb = sender as CheckBox;  
      var item = cb.DataContext;  
      list.SelectedItem = item;  
    }  
    private void btn_Click(object sender, RoutedEventArgs e)  
    {  
      string constr = @"constr ";  
      SqlConnection con = new SqlConnection(constr);  
      con.Open();  
      foreach (Data item in list.Items)  
      {  
        if (item.IsChecked == true)  
        {  
          string sqlinsert = "INSERT INTO [dbo].[Table]([Id],[Name])VALUES(@Id,@Name)";  
          SqlCommand cmd = new SqlCommand(sqlinsert, con);  
          cmd.Parameters.AddWithValue("@Name", item.Name);  
          cmd.Parameters.AddWithValue("@Id", item.Id);  
          SqlCommand cmdCount = new SqlCommand("SELECT count(*) from [dbo].[Table] WHERE Id = @Id", con);  
          cmdCount.Parameters.AddWithValue("@Id", item.Id);  
          int count = (int)cmdCount.ExecuteScalar();  
          if (count == 0)  
          {  
            cmd.ExecuteNonQuery();  
            MessageBox.Show("item saved");  
          }  
          else  
          {  
            MessageBox.Show("item exists");  
          }  
        }  
      }  
      con.Close();  
    }  
  }  
  public class ViewModel : INotifyPropertyChanged  
  {  
    public ViewModel()  
    {  
      Items = new ObservableCollection<Data>();  
      Items.Add(new Data() { Id= 1, Name = "Beijing", IsChecked = false });  
      Items.Add(new Data() { Id = 2, Name = "Shanghai", IsChecked = false });  
      Items.Add(new Data() { Id = 3, Name = "Guangdong", IsChecked = true });  
      Items.Add(new Data() { Id = 4, Name = "Beijing", IsChecked = true });  
    }  
    private ObservableCollection<Data> _items;  
    public ObservableCollection<Data> Items  
    {  
      get => _items;  
      set  
      {  
        _items = value;  
        OnPropertyChanged(nameof(Items));  
      }  
    }  
    public event PropertyChangedEventHandler PropertyChanged;  
    protected virtual void OnPropertyChanged(string propertyName)  
    {  
      PropertyChanged?.Invoke(this, new PropertyChangedEventArgs(propertyName));  
    }  
  }  
  public class Data  
  {  
    public int Id { get;set;}  
    public string Name { get; set; }  
    public bool IsChecked { get; set; }  
    public override string ToString() => Name;  
  }  
}  

The result:

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To start off, the following code is meant to be to learn from, not to match your current requirements. I use a database but the code can work without a database. So with that you may or may not want to continue, it's your choice.

• Frontend project
• Backend project
• Sql script
• INotifyPropertyChanged can be removed for Country.Name and the code still picks up on changes

At the moment nothing is displayed in the ListBox maybe because it is filled with TextBoxes.

<ListBox x:Name="AutoStrucNames" Grid.Row="5" Grid.Column="100" Height="600" Margin="80,20,0,40" Width="160" SelectionMode="Single" HorizontalAlignment="Left" Grid.ColumnSpan="10" Background="PowderBlue"
DataContext="{Binding ElementName=main}" ItemsSource="{Binding AutoNames}" >
<ListBox.ItemTemplate>
<DataTemplate>
<StackPanel Orientation="Horizontal">
<CheckBox x:Name="CheckAccepted" Margin="3" VerticalAlignment="Center" IsChecked="{Binding IsChecked, Mode=TwoWay}" />
<TextBox Text="{Binding AutoNames, UpdateSourceTrigger=PropertyChanged, Mode=TwoWay }"/>
</StackPanel>
</DataTemplate>
</ListBox.ItemTemplate>
</ListBox>

The strings that I have to display are in List <string> AutoNames.

I just try to display as follows:

AutoStrucNames.ItemsSource = (System.Collections.IEnumerable)AutoNames;

I see only the ListBox with many empty strings with a CheckBox . I feel frustrated.

You need to implement INotifyPropertyChanged if you want to subscribe to notifications that a property has changed. If you don't need to you don't have to implement this interface.

In the case of <TextBox Text="{Binding Value, UpdateSourceTrigger=PropertyChanged, Mode=TwoWay }"/> you don't need to implement INotifyPropertyChanged. It's easy to figure out, take the property and remove notifications, run the code and inspect edited items.