There are a few ways to remove the context menu and the triangle.
First easy way to remove the context menu is at the level of the DockingManager
<avalonDock:DockingManager x:Name="dockManager"
Grid.Row="1" AnchorableContextMenu="{x:Null}" DocumentContextMenu="{x:Null}">
or in you override style as is :
<Style TargetType="{x:Type avalonDock:DockingManager}">
<Setter Property="AnchorableContextMenu" Value="{x:Null}"/>
<Setter Property="DocumentContextMenu" Value="{x:Null}"/>
</Style>
or you override the full style to create your own style. Take one of the basice style that come with the control and remove or add your own part of the existing style.
As for the triangle you can add a visibility to the avalonDockControls:DropDownButton style as is :
<avalonDockControls:DropDownButton
x:Name="MenuDropDownButton"
Style="{StaticResource {x:Static ToolBar.ToggleButtonStyleKey}}"
Focusable="False"
VerticalAlignment="Center"
Grid.Column="1"
Visibility="Hidden">
I would suggest not to do it this way because you cannot override it later unless you bind it to a property.
But this work fine.