Do I have to Close() a SQLConnection before it gets disposed?

The using keyword will close the connection correctly so the extra call to Close is not required.

From the MSDN article on SQL Server Connection Pooling:

"We strongly recommend that you always close the connection when you are finished using it so that the connection will be returned to the pool. You can do this using either the Close or Dispose methods of the Connection object, or by opening all connections inside a using statement in C#"

The actual implementation of SqlConnection.Dispose using .NET Reflector is as follows:

// System.Data.SqlClient.SqlConnection.Dispose disassemble
protected override void Dispose(bool disposing)
{
    if (disposing)
    {
        this._userConnectionOptions = null;
        this._poolGroup = null;
        this.Close();
    }
    this.DisposeMe(disposing);
    base.Dispose(disposing);
}

Since you have a using block, the Dispose method of the SQLCommand will be called and it will close the connection:

// System.Data.SqlClient.SqlConnection.Dispose disassemble
protected override void Dispose(bool disposing)
{
    if (disposing)
    {
        this._userConnectionOptions = null;
        this._poolGroup = null;
        this.Close();
    }
    this.DisposeMe(disposing);
    base.Dispose(disposing);
}

Disassembly of SqlConnection from using .NET Reflector:

protected override void Dispose(bool disposing)
{
    if (disposing)
    {
        this._userConnectionOptions = null;
        this._poolGroup = null;
        this.Close();
    }

    this.DisposeMe(disposing);
    base.Dispose(disposing);
}

It calls Close() inside of Dispose()

Using Reflector, you can see that the Dispose method of SqlConnection actually does call Close();

protected override void Dispose(bool disposing)
{
    if (disposing)
    {
        this._userConnectionOptions = null;
        this._poolGroup = null;
        this.Close();
    }
    this.DisposeMe(disposing);
    base.Dispose(disposing);
}