Since this project will include documentation, I'm trying to build the docs. However, from the Makefile at scipy/doc it appears that python3.6 is required, whereas only python3.5 is available for my platform (Ubuntu 16.04 LTS). I generally try to avoid adding third-party repos unless absolutely required, so I am asking for help here. Is installing python3.6 the only solution, or will changing the PYVERS variable in the Makefile work?

I suspect 3.5 should work once you get it to use your Python binaries. I doubt we have anything that is Python-3.6 specific (though there very well be some Python 3-isms that will not work on 2.7).

Regarding the GSOC blog that needs to be set up, what is the preferred method/site/url for scipy? And what exactly should a blog post contain? I'm a bit unclear on those details.

You can choose any blogging platform you want. Usually people use it to give approximately weekly updates about progress, difficulties, successes, cool tricks, etc. See how it's mentioned here (and elsewhere in the student guide):

https://google.github.io/gsocguides/student/time-management-for-students

Eric