I think a spiral is a 2D construct, so what you describe is not really possible in 3D. Even the fact you don't know what it would look like should hint at that.
Hint: if you did manage to get the effect implemented in 3D, it wouldn't look very good. It would probably just look like a sphere with smudges going up or right (depending on which way Z points) was super-imposed onto your background (kind'a like the two rotating spirals here, just without the rotation: http://en.wikipedia.org/wiki/Spiral#Three-dimensional_spirals).
Also, a cubemap is still a 2D map, just mapped to a cube. The Z coordinate in this case is not used as 3D depth, it is only used to interpolate between the colors on opposite sides of the cube. Because of that interpolation, you can't really achieve a 3D effect with it. For a 3D effect you would need to be able to use your own algorithm for computing the color based on depth.
What you could do instead is map the 2D coordinates already used in your algorithm to 2D polar coordinates on a sphere (you said you already have this), and apply the 2D effect as though it was "stamped" on the surface of the sphere. At least that's what real cloud maps look like to me. The swirls caused by hurricanes only produce horizontal cloud spirals, across the surface of the Earth - there is no vertical spiral. I assume this is what you want?
I'm targetting DX11.I thought those SDKs are, like, downwards compatible
The SDKs are not backwards compatible. At least not the debug layer DLLs, which don't always have different names under different SDKs.
You also should not expect it to work if you install ALL of the SDKs, especially if you install them in the reverse order. Usually, the SDKs are designed so that a newer version can be installed over an older one, without breaking the older SDK's functionality (though this is not always the case).
If you did install all the SDKs successfully (or one by one), you should have (at one point had) the correct debug layer DLL installed, and your project should have worked. I forgot to mention DirectX 11.1 before, which has it's own debug layer DLL - are you targeting 11.1 maybe?
You could also try to install the DirectX redistributable you are targetting. It might be that Windows 10 SDK only supports DirectX12 by default, and the older DirectX versions are not included yet.
For DirectX11, you need to have D3D11SDKLayers.dll in the windows\system32 folder, or in your program's folder.
EDIT: I just tested my own DirectX 11 project in WIn10 with debugging enabled, and even tohugh I had D3D11SDKLayers.dll in the program folder, it seems that the directx runtime from Windows 10 is trying to load the debug dll from DirectX 12. DbgView shows this message:
 D3D11CreateDevice: Flags (0x
 ) were specified which require the D3D11 SDK Layers for Windows 10, but they are not present on the system.
 These flags must be removed, or the Windows 10 SDK must be installed.
 Flags include:
I installed the Graphics Tools feature and now it works... I thought this was because of the wrong SDK being installed. Sorry.