SPOILER (click to view)

Rika is basically Oyashiro. She's trying to get out of dying each time by resetting the time back and trying again, and each of the chapters or arcs are actually new resets. The invisible person Rika is talking to in Kai is Hanyu. She appeared in the Higurashi Daybreak game, and according to the story she can only be seen by Rika, which is why it only shows Rika talking to a shadow.

And about the first series, there are numerous explanations for each of the events, but simply put, each of the chains just shows another person going insane because of them seeing things that aren't happening (according to the original game and manga). For example, in the first arc, Keiichi thinks he sees Rena and Shion going insane, but in fact he's imagining it. He even imagines the needle in the desert, which is why he can't find it later. Oishii's information to Keichii only feeds his paranoia, to the point when Rena and Shion capture him and play the 'punishment game' they pull out a sharpie like before to draw on his face but he's lost his touch with reality so much he thinks he sees a syringe. It's similar for the other arcs as well. Oh, and the syringe Rika carries in Shion's arc, is actually a drug to try to help cure Shion of her insanity, but since shes too weak she gets beaten by Shion. Having the drug injected into her doesn't have any effect, but having lived through Shion going insane before in an earlier reset, she decides to kill herself instead of being tortured painfully to death.