WPF, Whats with this weird whitespace in Tooltip/Border

That rectangle comes from the ControlTemplate of the ToolTip control wich will contain whatever you set in Control.ToolTip. You have to override it:

<Style TargetType="ToolTip">
    <Setter Property="Template">
        <Setter.Value>
            <ControlTemplate TargetType="ToolTip">
                <ContentPresenter />
            </ControlTemplate>
        </Setter.Value>
    </Setter>
</Style>

Have you tried reducing the padding of the tooltip directly:

<ToolTip Padding="0" >
...
</Tooltip>