How can I have a ListBox auto-scroll when a new item is added?

You can extend the behavior of the ListBox by using attached properties. In your case I would define an attached property called ScrollOnNewItem that when set to true hooks into the INotifyCollectionChanged events of the list box items source and upon detecting a new item, scrolls the list box to it.

Example:

class ListBoxBehavior
{
    static readonly Dictionary<ListBox, Capture> Associations =
           new Dictionary<ListBox, Capture>();

    public static bool GetScrollOnNewItem(DependencyObject obj)
    {
        return (bool)obj.GetValue(ScrollOnNewItemProperty);
    }

    public static void SetScrollOnNewItem(DependencyObject obj, bool value)
    {
        obj.SetValue(ScrollOnNewItemProperty, value);
    }

    public static readonly DependencyProperty ScrollOnNewItemProperty =
        DependencyProperty.RegisterAttached(
            "ScrollOnNewItem",
            typeof(bool),
            typeof(ListBoxBehavior),
            new UIPropertyMetadata(false, OnScrollOnNewItemChanged));

    public static void OnScrollOnNewItemChanged(
        DependencyObject d,
        DependencyPropertyChangedEventArgs e)
    {
        var listBox = d as ListBox;
        if (listBox == null) return;
        bool oldValue = (bool)e.OldValue, newValue = (bool)e.NewValue;
        if (newValue == oldValue) return;
        if (newValue)
        {
            listBox.Loaded += ListBox_Loaded;
            listBox.Unloaded += ListBox_Unloaded;
            var itemsSourcePropertyDescriptor = TypeDescriptor.GetProperties(listBox)["ItemsSource"];
            itemsSourcePropertyDescriptor.AddValueChanged(listBox, ListBox_ItemsSourceChanged);
        }
        else
        {
            listBox.Loaded -= ListBox_Loaded;
            listBox.Unloaded -= ListBox_Unloaded;
            if (Associations.ContainsKey(listBox))
                Associations[listBox].Dispose();
            var itemsSourcePropertyDescriptor = TypeDescriptor.GetProperties(listBox)["ItemsSource"];
            itemsSourcePropertyDescriptor.RemoveValueChanged(listBox, ListBox_ItemsSourceChanged);
        }
    }

    private static void ListBox_ItemsSourceChanged(object sender, EventArgs e)
    {
        var listBox = (ListBox)sender;
        if (Associations.ContainsKey(listBox))
            Associations[listBox].Dispose();
        Associations[listBox] = new Capture(listBox);
    }

    static void ListBox_Unloaded(object sender, RoutedEventArgs e)
    {
        var listBox = (ListBox)sender;
        if (Associations.ContainsKey(listBox))
            Associations[listBox].Dispose();
        listBox.Unloaded -= ListBox_Unloaded;
    }

    static void ListBox_Loaded(object sender, RoutedEventArgs e)
    {
        var listBox = (ListBox)sender;
        var incc = listBox.Items as INotifyCollectionChanged;
        if (incc == null) return;
        listBox.Loaded -= ListBox_Loaded;
        Associations[listBox] = new Capture(listBox);
    }

    class Capture : IDisposable
    {
        private readonly ListBox listBox;
        private readonly INotifyCollectionChanged incc;

        public Capture(ListBox listBox)
        {
            this.listBox = listBox;
            incc = listBox.ItemsSource as INotifyCollectionChanged;
            if (incc != null)
            {
                incc.CollectionChanged += incc_CollectionChanged;
            }
        }

        void incc_CollectionChanged(object sender, NotifyCollectionChangedEventArgs e)
        {
            if (e.Action == NotifyCollectionChangedAction.Add)
            {
                listBox.ScrollIntoView(e.NewItems[0]);
                listBox.SelectedItem = e.NewItems[0];
            }
        }

        public void Dispose()
        {
            if (incc != null)
                incc.CollectionChanged -= incc_CollectionChanged;
        }
    }
}

Usage:

<ListBox ItemsSource="{Binding SourceCollection}" 
         lb:ListBoxBehavior.ScrollOnNewItem="true"/>

UPDATE As per Andrej’s suggestion in the comments below, I added hooks to detect a change in the ItemsSource of the ListBox.