Upgrading comment to an answer here...
1) To set the UserControl as its own DataContext just do a this.DataContext=this;
in the constructor. You'll almost never want to do this for anything other than development/testing though because if the person using your control sets the DataContext to something else (i.e. to the data they actually want to display) then it will break all your bindings.
2) It will normally look something like this:
Binding="{Binding RelativeSource={RelativeSource Mode=FindAncestor, AncestorType=your_namespace:ucLCGPulseWindowMini}, Path=IsShowing}"
3) If you don't want to use FindAncestor then you give the user control a name and refer to it directly:
<UserControl
x:Class="Your.NameSpace.ucLCGPulseWindowMini"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
x:Name="myControl"
d:DesignHeight="300" d:DesignWidth="300"
>
... Binding="{Binding ElementName=myControl, Path=IsShowing}" ...