OK, so I have been working with Infragistics support and delving into the depths of their documentation and I found what I was looking for.
Evidently, when a pane is floating routed events will not bubble up to the main window and styles defined in the main window will not apply to elements in a floating pane (this includes data templates).
For the styles and data templates, there are two possible solutions:
- As I mentioned in the question, you can place the styles in App.xaml.
- You can handle the ToolWindowLoaded to set styles or attach event handlers for the floating pane. In conjunction with ToolWindowLoaded, you can handle the ToolWindowUnloaded event to remove those event handlers and styles when the pane is no longer floating.