Q:

How can I install and use build tools v120 (2013) on Visual Studio 2015 from a Windows 7 machine? 

I need to build several C++ projects (originally developed with VS 2013) that require v120 build tools.  I can only use VS2015 (due to licensing and other corporate formalities I cannot install VS2013).  
As per multiple similar SO questions, such as this question or this question, it appears my only options are to:

Install VS2013 - (NOT POSSIBLE)
Use VS2015 installer to install Windows 8.1 and Windows Phone 8.0/8.1 - also NOT POSSIBLE since the machine I am using is running Windows 7 and Windows 8.1 and Windows Phone 8.0/8.1 just doesn't appear in the installer. 
Retarget the projects to build using v140 - again NOT POSSIBLE as this will effectively render the projects unusable by my colleagues who also work with these projects. 

Isn't there any standalone installer that I can use to install those build tools, or any other option I can try? 

A:

Isn't there any standalone installer that I can use to install those build tools, or any other option I can try? 

I am afraid there is no such standalone installer that you can use to install the build tools v120 on the windows 7.  Just as you found, the option Windows 8.1 and Windows Phone 8.0/8.1 Tools could not be installed on the windows 7. 
But, there is a workaround to resolve this issue, install the Visual Studio 2013 express instead of Visual Studio 2013. 
You can download it here. 

Visual Studio 2013 express will resolve your licensing restriction, but I am not sure if it match your corporate formalities.  If not, I think you can consider stijn`s suggestion "use $(DefaultPlatformToolset) as toolset instead of hardcoding the toolset to v140"
Hope this helps.