What do you mean by backwards compatible? If you mean modern windows OS's can still run applications that use older versions of directx then yes but truthfully in that case its windows that is backwards compatible not directx. The latest version of DX (12) is significantly different than previous versions. As far as how long DX will be around for I'd say as long as windows is around.

This is a tricky question:

If you have DirectX 9/10/11 installed, then you have the older versions still installed. This means e.g. a DirectX 6 application will still work.

However starting Windows Vista, Retained Mode was deprecated (which was used optionally by DirectX 3.0, DX 5 and DX 6 applications). Apps using Retained Mode needed lots of workarounds and not all of them may work on Vista+. But if they didn't use Retained Mode, then they should work (in theory) even on Windows 10.

Furthermore, vendors rarely test their latest GPUs against such old DIrectX versions, so it's possible it won't work correctly on certain GPU / driver combinations (specially if the app didn't actually follow the DX spec rigorously, which was very common). Though most of the time these programs would often had trouble running in D3D mode even back in their time, developers just blamed on the GPU, the driver, or Windows 98. Examples out of my mind are Grim Fandango, Final Fantasy VIII, Startopia (enabling HW TnL would result in black floor).

You may get it to work using RGB software rasterizer or via virtual machine with custom drivers.

Furthermore, before DX10, DirectX had a 3-version backwards compatibility. That meant a GPU designed for DirectX 5.0 would still work on an app using DirectX8, but not on DirectX9.

Starting DX10, Microsoft introduced Feature Levels. An app would only run if it supported the required minimum feature levels by the GPU. So an app using DirectX 10 Feature Level 9.1 would probably run on a lot of GPUs; while still using the DX11/10 API.

The same applies to DirectDraw. In the beginning they were separate, one for 2D the other 3D.

Beginning DirectX 8, DDraw began a slow deprecation (moving towards doing everything in D3D) until it was completely phased out in DirectX10.

DDraw could have bugs or not, just like D3D applications. But it's more likely to have fewer bugs. A major advantage is that on Windows XP DDraw acceleration can be turned off as CPUs should be fast enough the kind of work we supplied in the 90's to DDraw; and in Windows Vista+ it can be turned off via the DirectX Control Panel switch or reg keys (though not beginners).