None. LLVM is a compiler toolchain. Basically it takes care of the back-end of your compiler. .NET is a framework (native libraries + CLR) usually associated with C# ecosystem.
I forgot about Blazor and all that WebAssembly stuff... Thank you, I think this is why open-dotnet maintains a fork of LLVM.
They don't really have anything to do with each other. (There was a project for a LLVM-based compiler for the .NET ecosystem, but as far as I know that never went anywhere)
Yes. Because dotnet is inherently JIT (e.g. linq expression tree and runtime code weaving and etc.) and you can't do that with llvm. So it was a dead end effort.