Is it possible to template a ListBoxItem so that in the visual tree the ContentPresenter is left out and I have direcctly the data template underneath the ListBoxItem?
You may replace the ListBoxItem Style by one that does not have a ContentPresenter, like this very basic one:
<ListBox>
<ListBox.ItemContainerStyle>
<Style TargetType="ListBoxItem">
<Setter Property="Template">
<Setter.Value>
<ControlTemplate TargetType="ListBoxItem">
<TextBlock Text="{Binding}"/>
</ControlTemplate>
</Setter.Value>
</Setter>
</Style>
</ListBox.ItemContainerStyle>
</ListBox>
For a more realistic Style you may copy the default ListBoxItem Style from here and replace the ContentPresenter by something that fits your needs.