In the first episode, Eiko asks Ika how she plans to deal with the earth's military, pointing out that they have guns and missiles. Not knowing what these are, Ika mishears the word "missiles" as "miso soup". Considering that Ika lived under the sea for her entire life, one wonders how she even knows what soup is.
In Shinryaku! Ika Musume, Ayumi Tokita wears an oversized mascot head on a near-constant basis - partly to promote her father's beachside restaurant, but partly because of crippling insecurity in dealing with others. The first thing her father does when he meets Ika Musume is offer to trade his mascot to another restaurant in exchange for Ika, the "superior" mascot. Though it's never brought up directly, it's not hard to see where Ayumi got her insecurity.
It's been shown on a few occasions that when Ika Musume suffers an injury, black liquid, most likely ink, will drip from her, usually from her nose. If her blood is ink, that would mean every time she produces it for squid ink spaghetti, she's puking up her own blood. This makes the scene where Chizuru forces her to do so to the point she withers even more uncomfortable to watch.
Fridge Brilliance: That said, it's also possible that she's doing what squids naturally do when they feel threatened, squirting the ink out to distract the "threat".
Less Horror and more 'Fridge Tearjerker'. All those Mini Ika shorts - you know, the off the wall, funny slice of life dreams. Well, all evidence points to each episode being in the same universe as the others, which means All those moments will eventually lead to Eiko dying alone in the hospital, with Mini Ika desperately trying to wake her. Damn it!
On the same note, if what the doctor said about Mini Ika living longer then humans applies to regular Ika Musume, that means she too will live longer then any of her friends.
When Takeru's squid hat blows off, squid girl acts as though her new squid soldier had died WHILE acknowledging that Takeru was still alive. We can safely infer that Squid Girl's hat might be in fact be a puppeteer parasite.