Direct3D 10.1 is interoperable with the Windows GDI, but there is very little explicit documentation on the matter. MSDN has a diagram with Direct2D at the center of the interoperability web, but we have to rely on a DirectX blog post for the complete interoperability diagram. In particular, note that the latter diagram has a path from Direct3D10.1 to GDI via DXGI 1.1.

MSDN gives the impression that this interoperability is simple: when calling CreateTexture2D, there is a nice flag called D3D10_RESOURCE_MISC_GDI_COMPATIBLE, the documentation for which says that after enabling the flag, the resulting texture can be cast to an IDXGISurface1 and have GetDC called on it. A similar flag called DXGI_SWAP_CHAIN_FLAG_GDI_COMPATIBLE exists when creating a swap chains rather than textures. Unfortunately, if you try to naively use one of these flags, resource creation might well fail with E_INVALIDARG. The reason for this failure is stated on GetDC, but really needs to be more prominent:

The format for the surface or swap chain must be DXGI_FORMAT_B8G8R8A8_UNORM_SRGB or DXGI_FORMAT_B8G8R8A8_UNORM.

If this constraint isn't satisfied, then it isn't GetDC which will fail, but the resource creation itself.