The actual deity of the town of Kannagi is the goddess Kannon.
According to the Beach Episode chapter of the manga, it is possible for a god to live as separate entities while also still being a god, as evidenced by the four humans who were also the god of the town that the art club visited. It might be that Nagi is actually an entity of Kannon's who forgot her original name. Kannon is a motherly goddess of compassion and mercy who tries to salvage as many souls as she can, helping them to reach enlightenment. The goddess personality that Nagi assumes from time to time seems to be in accord with this behavior, absorbing the dead kittens into her body and preventing the many suicides from taking place using her sacred tree. The decendants of the town itself have forgotten that their shrine was actually one of the many dedicated to the goddess Kannon because they had fallen out of practice of offering her worship, and their damaged records of the name of the god of the shrine don't help them to remember this. Their lack of worship is also why Nagi is so powerless when she is first revived. Interestingly, during the persecution of Christianity in the Tokugawa Era, some Japanese Christians created statues of the Virgin Mary disguised as Kannon; while these statues looked like Kannon, they were worshipped as Maria Kannon (the Virgin Mary in the guise of Kannon), creating a nice parallel to Zange in the story as well. Akiba is a Beta Couple with Takako and Shino.
The reason why Takako and Shino aren't part of Jin's harem, is because they're after Akiba. The trio are shown to regularly hangout together whenever Jin is dealing with Tsugumi, Zange, and Nagi. Takako especially likes to tease Akiba and push his buttons, while Shino pulls a Straight Man whenever he goes too far. The three also spend a lot of time together in the club room. We only see parts of their daily antics whenever Jin is present, but it's clear they continue the fun before and after Jin's there.