These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Shinka is arguably more obsessed than the rest of the cast when it comes to creating a separate identity for herself. Over the course of the anime series, she signs up as the class representative, joins several different clubs, dyes her hair, and attempts to become president of the student council. All of this is done in order to establish herself as something other than a chuuni.
Broken Base: Fan opinion on episode 7 is split. Some feel that the drama is unnecessary, and that Rikka doesn't need a psychological reason to behave the way she does. Others like the depth and development that it brings her character, as well as the show itself.
In episode 11, Yuuta became one, thanks to a combination of forcing Rikka to give up her chuunibyou ways (to the point of trying to forcably rip the eyepatch off of her) and seeming completely oblivious to how miserable she is when she does give it up.
Except he's not oblivious to it at all. He knows how much pain Rikka's going through and is even angsting about how guilty he feels about his actions. He just isn't willing to let himself reassure her because he feels it would be irresponsible and go back on his previous actions.
The finale of the second season has already become this. A large chunk of the audience hates that they spend the whole season building up to a Big Damn Kiss only to have it NOT happen at the very end. Typical for Kyoto Animation, but reeks of a cop-out.
So much so that they end up accidently kissing in the bonus OVA.
Dekomori's response to Rikka saying she's in love with Yuuta in episode 10 can be seen as this, completely with Murder the Hypotenuse (she snapped out of it fast and helped to give her "master" confidence to confess to Yuuta).
I think it's safe to say that pretty much every character in this series is moe to some degree.
Rooting for the Empire: Quite a lot of people who watched the anime became desillusioned about the main pair and expected something more deconstructive going on, like parts of the Light Novel. Consequently they took sides with Yuuta pre-12th episode and Touka rather than how the show eventually turned out. Others simply side with Touka on principle of being able to "defeat" the heroes regularly with but a ladle.
Tear Jerker: The end of episode 7 and the beginning of episode 8.
After Rikka is hit with the fact that her father is dead, we can't see how she is going to react because the half of her face we are shown is obscured by her eyepatch. Then a tear rolls down.
Rikka finally accepting her father's death.
Rikka: Goodbye, father.
What Do You Mean, It's Not for Kids?: Despite being very lighthearted and not very violent, both of the anime adaptations are rated "TV-MA" when streamed on Hulu. This is highly unusual, because on IMDB it is rated "TV-G". While it may not be the best show for young children to watch, it certainly doesn't warrant a "mature" label. "TV-14", is probably the ideal rating.