Tamai is Kokonotsu's mother
Yo has apparently known her for years, is able to argue with her, and then is comfortable enough with her to ask her to go drinking, and she's comfortable enough to accept. Later in season 2, he's willing to take shelter at her house during a storm.- Jossed. Yo talks to Tamai at one point about his wife leaving him in the manga.
A lot of Hotaru's eccentricities come from being a Lonely Rich Kid
She wants friends, but has no idea of how to go about getting some. Kokonotsu met Hotaru when he was young, but doesn't remember the encounter.
Which is why her likeness shows up in his manga. Hotaru was the mystery woman who ordered the Monjiro Squid for Yo.
She probably heard he was in the hospital and ordered it for him. Hotaru has some -very- interesting relatives.
Judging from the Namaiki beer episode, Hotaru has probably seen a lot of that stuff happen at family gatherings. Saya and To's family are well off.
We never see their parents or know what happened to them. However, they do have a cafe which seems to provide for all their material needs, despite not seeming to get very many customers. Its very possible that Saya and To have a trust fund that they're living off of.