The reason why Katsuragi's teacher has a choker on? She's partnered up with an undercover devil! In fact, the majority of keen — or just experienced — readers suspected her choker to be the guillotine collar from earlier on in the series. But after the picture of her in her high school years revealed she didn't have the choker then, it was simply regarded as a fashion accessory. It is a simple, yet cunning, ruse set up by Wakaki-sensei to confuse the most savvy readers. Or... is the choker really a fashion accessory, though she is supposedly an elite devil?
By the same token as the above, Akari's scarf never comes off.
Keima's plan in chapter 122? It's Real Life Capturing God Mode.
Look at Tsukiyo's poker card. In the background is Luna, apparently standing up. Something that a normal doll wouldn't be able to do.
Apollo and her host Kanon. It is apt that Kanon has Apollo for a Goddess, seeing as how Apollo was patron god of the Sun, the brightest object in our sky, as well as healing and music. Also Mercury and her host Ayumi. Though Ayumi has always been an extremely fast runner, it is only appropriate that she have Mercury. When Chihiro asked Ayumi how the latter found out about her confessing to Keima, she said that she heard it from whispers in the wind. Note that Mercury is the Roman equivalent of Hermes, the fastest among the gods, and their messenger.
Furthermore, one of Mercury's traits in mythology are his winged shoes/talaria which flew him around at fast speeds, as befits of his status as the messenger to the gods. Considering the fact that Ayumi is on the track team (and therefore does a lot of running), it's much more likely for the last conquest target to be her rather than Chihiro.
According to Haqua, human hosts of weak runaway spirits can manifest the spirit's powers; thus explaining the reason why Kanon turned invisible whenever she felt she's being ignored or Kusunoki having a Literal Split Personality. Then we remember Shiori, who managed to memorize every book she read in the library, which Keima attributed to the runaway spirit possessing her at that time. Fast forward to several hundred chapters after, we find out she is the host of the goddess Minerva. Connect two and two together, and we get Shiori's ability stems not from the runaway spirit possessing her, but from the goddess inside her. It also helps that Minerva is the roman goddess of wisdom, so she would naturally be drawn to places of knowledge, hence why Shiori's usual place was at the library.
Chihiro has quite a lot of this, despite being the average girl among Keima's conquest targets. Mainly because she is actually one of the few capture targets who fell in love with Keima's real personality, making her more unique than she may realize.
When Keima was teaching her how to successfully confess to the guy she was after, he made her spill a cup over that guy. What is the first thing she does during the recapture arcs? Spill a cup over Keima. Twice. How does she make his work really difficult for the second time many chapters later? Confess to him without Keima actually reaching his intended "ending".
Chihiro was also the only girl who openly acted against his conquest, saying that she wasn't someone he would pay attention to. Just like nobody she ever confessed to paid attention to her during her conquest and before it. That's because she was in love with Keima for some time (even before her conquest happened), and started confessing to people once she realized she probably won't be able to get Keima. Or the fact that she has, at some point, fallen for the least popular guy in their school. (She did state that she fell for him one Spring time ago, way before the events of her conquest.)
The fact that Chihiro loved Keima before the conquest makes even more sense after you realize that her conquest didn't involve Keima putting on a persona. Since he was attempting to help her confess to another guy, Keima was still acting like well... Keima. Just by acting like himself, Keima unintentionally conquered Chihiro.
Chihiro not having a goddess makes sense. She's always been labeled as the average girl. Even though she was suspected of having a goddess, as it turns out, she was still normal.
In the first chapter of Chihiro's capture arc, Elsea's spirit sensor goes off right after Keima loudly claims that there was no way he would be concerned with someone like Chihiro. Considering that she was nearby, it's possible that a hole opened in her heart because she heard him.
When cleaning the classroom with Keima, Chihiro asks mockingly if he's ever been on a date or kissed a girl in real life—and to her surprise, he blushes and says "Those didn't count." She then demands to know who it was. Because she wanted to know who her competition was.
Chihiro's flaw was wanting to be special, so she confessed to guys who looked special because if she was with them, she would feel special as well. She fell for Keima well before her capture because she knows Keima is special, although in an unconventional way, and getting to know him better during her capture arc merely reinforced her feelings.
The anime gives a subtle hint to Chihiro's possible romantic interest in Keima well before her capture. More specifically in episode two of season one; as Keima contemplates how many runaway spirits he has to capture, the camera moves until Chihiro is seen. Note that she was looking at Keima (at 12:09) before Elsie got her attention.
Keima constantly refers to reality as "a shitty game," not worth playing. However, we've seen his reaction to a shitty game—play it anyway, for the sake of the girls. His slow Character Development and feelings for the capture targets makes a lot more sense from that perspective. As he himself puts it, "There are no bad heroines, just bad games." In chapter 251 he actually lampshades this.
Elsie's introduction of herself as Keima's sister was at school, also conveniently being enrolled in the same class, but at that point, she has yet to meet Keima's mother, so how did she get enrolled? Nikaido-sensei, under chief Dokuro's orders, arranged things for her.
Diana can communicate with Tenri via telepathy, and also with others through Tenri's reflection via reflective surface, and lastly, by taking full control of Tenri's body. So how in the Real Life physics can Diana emit sound from reflective surfaces? Answer: She uses those as surface vibration speakers
The way human-devil partnerships are set up may seem random at first glance, as evidenced by Keima/Elsie and Haqua/Yukie. But when given a fair amount of thought, it makes perfect sense. The impatient Haqua, who wants to do things quickly, gets partnered with the middle aged Yukie, who takes things slowly but surely, to teach Haqua to take things slowly. The bubbly and friendly but highly inefficient Elsie gets Keima, the epitome of efficiency, but lacks social skills, is intended on training Elsie to be a more competent individual while at the same time, teach Keima to open up to the real world. Even the partnership of the dominating Demon Nora and her submissive human buddy Ryou makes sense, given their personalities. Whoever was setting these partnerships up must have intended them to be that way.
More on the buddy system. The reason devils are supposed to let their human partners deal with the spirits directly is so that if something goes wrong and the spirit is able to possess someone new, it ends up controlling a weak human instead of a magic-slinging devil with a Celestial Robe. Like what happened with Haqua.