When we install SQL Server it does not ask to install .Net Framework but all latest version of Windows have .Net installed by default. I recently did some programming (in C#) against SSIS engine and I saw following DLLs.

Now I could just refer this DLL in .Net project and start calling functionality from it… I did not needed to do any Inter-op. My understanding is this all are .Net DLLs SQL Server engine will also be using them so is it a .Net application?
I am looking for this information as I was wondering for Performance we go for native (Windows SDK) as .Net Frameworks SDK comes with overhead for Stability and definitely Enterprise RDBMS needs to show performance.
SQL Server 2005 and up have started supporting .NET object internally. You can reference .NET assemblies, etc.
You do need the .NET framework to install SQL Server.
However, the core database engine of SQL Server is still written in native code.