So I finished watching the first 3 seasons and I still don't fully understand some things.
1. This is really the only important piece that I really want to learn about. Frederica Bernkastel. What was her connection to the Higurashi world and Rika. I haven't watched Umineko yet so maybe that can clarify some things about her but why would she just appear in Kai for a few seconds with young Takano without the audience having any clue who she is and how she is important. Surely she plays some sort of important part in Higurashi, right?
2. Hanyu no longer being in the physical world after Matsuribayashi-hen. No one notices and there is no explanation?
3. In the first episode of Kai, Rena lives to be in her 30's yet she should have died from Hinamizawa Syndrome shortly after Rika's death.
Already you did something wrong. First of all, read the VN it explains it all.
Afterwards, if you want to know more about Bern, read the Umineko VN.