First, yes "Visual Studio Express 2012 for Windows Desktop" is the actual name of product. :)
If you want to be on the bleeding edge, you can use VS Express 2012 for Windows Desktop with Integration Studio 7.0.014 (and maybe others but I've only tested that version).
All you have to do is, in Integration Studio, go to Edit -> Options and change the .NET settings to:
Set ".NET Integrated Development Environment" to C:\Program Files\Microsoft Visual Studio 11.0\Common7\IDE\WDExpress.exe
Set ".NET Compiler" to C:\Windows\Microsoft.NET\Framework\v4.0.30319\MSBuild.exe (you don't have to do this, but you'll get a warning when you publish the extension)
Also, don't forget to set the "Target Framework" of all projects inside the solution to ".NET Framework 3.5"
And in case you want to download Visual Studio Express 2012 for Windows Desktop, here you go: https://www.microsoft.com/en-us/download/details.aspx?id=34673. The ISO file is about 600 MB, but you can use the much smaller web installer.