I'm not sure which C extensions do you mean, but Clang supports most of the same GNU extensions that GCC does. Indeed, it's one of the goals of Clang to be compatible with GCC whenever possible. However, there might be some things that are not (yet?) supported.

And the kernel can be compiled using LLVM and Clang, but not quite in all configurations. But compiling mainline kernel for x86 using LLVM was achieved some years ago.