In episode 7, Natsumi started to read what it look like a manga magazine, and she started reading it with others.
Does it come from a real manga? If it does, what manga is that?
This is the Monthly Comic Alive. It is a Japanese seinen manga magazine published by Media Factory.
Picture from comicvine
This episode here is the July 2013 issue or as a number 201309.
I am pretty sure the mange she is reading here is Omae o Otaku ni Shiteyaru kara, Ore o Riajuu ni Shitekure! (おまえをオタクにしてやるから、俺をリア充にしてくれ！)
The scene you see on the left page of the secound image is in the tenth episode.
Picture from Mangapark
The manga on the cover is Seikoku no Dragonar. It has received an anime adaptation just last season (Spring 2014).
The manga that she is reading inside is Omae o Otaku ni Shiteyaru kara, Ore o Riajuu ni Shitekure! (lit. I will turn you into an Otaku, so please turn me into a Riajuu).
The pages are from the end of chapter 10. Below is the 2-page spread for comparison. Note that the art is not even distorted (which is usually done to avoid copyright problem), which suggests that it has received an okay from the author.