The official Visual Studio "DirectX" templates are only for Windows Store apps for Windows 8.x and Windows phone 8.x. They do not support Win32 desktop apps.
In order to develop Windows Store apps for Windows 8.1 or Windows phone 8.1 apps, you need:
- VS 2013 (any edition except Express for Web and Express for Windows Desktop)
- A Windows 8.1 system
In order to develop Windows Store apps for Windows 8.0 or Windows phone 8.0 apps, you need:
- VS 2012 or VS 2013 (any edition except Express for Web and Express for Windows Desktop)
- A Windows 8.0 or Windows 8.1 system
VS 2013 only supports Win32 desktop development when hosted on a Windows 7 system. You can certainly develop a DirectX app on such a system as a Win32 desktop app, but there's no built-in magic template to get you started. You create a standard Win32 desktop app, then add support for Direct3D from there. See Direct3D Win32 tutorial.
PS: The Windows 8.1 SDK is included as part of VS 2013 Express for Windows Desktop, VS 2013 Pro+, VS 2013 Community. You can download it as a standalone as you did as well.