1.I thought it was obvious that Juugo's clan had some relation to the juubi's power (his feats make it relatively clear, imo). I had just figured that it was a kekkai genkai unique to Juugo's clan based off of the Juubi. Just like how the kekkai genkai of the sharingan and the rinnegan that are present on the Juubi are present in the Uchiha clan.
Also in addition to what you said in your theory (atleast i didn't see this noted). like the juubi, juugo demonstrates the ability to heal extremely harsh wounds instantly. He heals sasuke near instantly from a fatal wound. The Juubi healed Obito from being cut in half, instantly.
2. Isn't everyone born with chakra somehow linked to the sage of the six paths? I'm pretty sure it is, since Madara had made a specific note that The so6p was born with chakra in his body (meaning everyone else at this time wasn't). How connected the link is I'm unsure. It is possible that his son who was born with his body, also had this ability and at some point in time the ability of natural energy absorption and the Senju's ability of naturally strong chakra diverged into seperate clans (1 ability passed onto two different children like what the sage did). If you want to go with that theory, Juugo's body in concept could be the sage's body, definitely a step towards becoming a new so6p.
3. I'm not going to say that is his clan symbol for sure, but it is a likely candidate for what they would choose as their symbol (since it is a part of their abilities). I didn't pay m uch attention to this at all. I didn't really care what Juugo's clan symbol was, but good find. Kinda looks like a rinnegan with fewer circles.
4. That would be interesting if it did, I wouldn't be particularly surprised. But I think it's a relic left to the uchiha by the sage and has special properties, one of those special properties could be what you said (The uchiha are good at hiding things (stone monument) so the being left by the sage wouldn't be hard to hide). Though I don't think it does use natural energy as a medium, and is just Madara's chakra flowing through it to make it work. The symbols are just symbols of the uchiha clan (sharingan). I wouldn't relate it to the heaven seal in my thought process, but that's just me. Naturally energy could be being used as a medium but I havn't seen any good proof to suggest it wasn't just normal chakra. Also as we've seen when Juubi embues someone or an object(susano) with natural energy, they get markings and change clearly in color, this did not happen with the fan. When Naruto uses natural energy his pupils change shape and gains color, i would figure if madara was using natural energy on the fan, something about it would change.
