To use TeeChart.Direct2D do I need to install SlimDX as well or is all that encapsulated within TeeChart.Direct2D.dll?
Yes you need install SlimDX, as explain in this doc
that I think can help you. Moreover, there are a good news for you, so in this document told, ClickOnce is installed automatically, when you select option to include SlimDX.
If TeeChart.Direct2D.dll is installed with my application on a non-compatible OS.
]a) Does the application throw an exception?
It doesn't throw an exception, only doesn't allow install SlimDX, and how consequence, doesn’t install Direct2D.
b) Does the application run and TeeChart.Direct2D does nothing. All charting/drawing occur as if TeeChart.Direct2D is not present?
You must make, an application where is used TeeChartDirect2D for O.S support SlimDX and the other application where is used TeeChart.Net for O.S doesn't support SlimDX.
I hope will helps.