You never need to install Visual Studio. For your information, not only you can run applications without Visual Studio, but you don't need it to build your solutions. The C# and VB.NET compilers are supplied with (freely re-distributable) .NET Framework, which is what you need.
As to SQL server, it depends. Do you know that it's based on the
client-server technology? So, if your application works with the remote server, it is, well… is already installed. In other cases, you need to provide it, too. How about SQL Server Express? It is also freely downloadable and distributable:
http://en.wikipedia.org/wiki/SQL_Server_Express[
^].
—SA