Unhandled C# Exceptions
Not unless you're dealing with unmanaged resources like raw memory access. (In which case the Exception isn't likely the cause)
.NET Exceptions will be caught by the runtime environment. You can crash your PROGRAM but not the OS.
In my experience the only things that cause BSOD's are bad drivers, and hardware errors. (This of course is not an extensive list of the causes of BSOD, but that's the first place I look.)
The only causes for a BSOD are hardware error, OS bug, or driver bug. .NET code cannot ever cause a BSOD.
No; ideally, there should be no way to cause a BSOD from C#.
Show us the code that is causing it and we'll see if we can help.