If there is any other ability for the sharingan then it should bear the name of a shinto god or some other deity.
Tsukuyomi, Susano, Amaterasu, Izanami, Izanagi, Kagutsuchi.
Maybe Hachiman the god of war.
Hachiman is a tutelary deity commonly associated with the dove, an avian creature. The epithet of Sasuke's organization was Taka, the literal translation of the word being "hawk". Interestingly enough, hawks and doves both hold an affiliation to conflict. The hawk represents the advocacy of strength and forcefulness. Taka disregarded international authority and trespassed on foreign territory numerous times throughout its existence. Not only that, they would rely on the strength of their abilities to achieve their ends.
What's also interesting is that the object of reverence for Hachiman is a bow. Sasuke's Susanoo incorporates a bow into its design. Although a shintai is really just a conduit for worship and not an actual characteristic of the deity.
If anyone was going to acquire a technique centred around Hachiman then it would be Sasuke.
Also, I believe that "Kamuy" is the Ainu term for their supernatural entities. Apasam Kamuy, Moshirikara Kamuy, Yushkep Kamuy... The Ainu are the indigenous populations that reside in Japan and a kamuy holds many similarities with the Japanese supernatural entities we commonly associate with the country. Now correct me if I'm wrong but I don't believe that there is any Shinto deity or collection of entities that have the name "Kamui". It's just a synonymous label referring to the authority and and strength to otherworldly beings in general.
If the above is accurate then Tobi and Kakashi's techniques would be an exception to the unspoken "MS techniques must have the appellation of a specific Shinto god or group of gods" rule that this manga is known for. It's related to deities, but the origins of the term "Kamui" are much more ambiguous in nature than any other mythological influences that Kishi has been inspired by.
As for what Madara's trump card is, well, that could be anything. Edo Rikudou Sennin, tengu warrior Susanoo, Black Zetsu (one can only hope), or the unspecified 4th eye technique. If it's a doujutsu ability then it will probably be grounded in either creation or Takamagahara mythology as those seem to be the two most utilised fields of Japanese supernaturalism when addressing the Mangekyou Sharingan.
On a side-note, there's a few pieces of information regarding these deities that I never understood until now. Kotoamatsukami is a collective designation of five deities in Kojiki, a creationist chronicle detailing the formation of the Japanese islands. Well apparently these five deities were formed in solitude and chose to secrete themselves. Isn't that reminiscent of Itachi concealing KA inside of Naruto only to have it appear nearly 200 chapters later when they encountered each other once again?
I also think that the Land of Waves arc may have been inspired by the Hare of Inaba myth.
Tazuna represents the hare, the deceiver who instigates a conflict with another person or group in the hopes of being able to journey from an island to the mainland. Gato and his organization could be interpreted as the the personification of the sharks. Tazuna opposed the pretences of the group for their oppression of the country and sought the assistance of others. The clan of hares symbolises the persecuted and oppressed people of the Land of Waves and the clan of sharks represents Gato and his criminal organization.
Okuninushi and his group would be Naruto and Team 7. Tazuna discovered them and requested their assistance. At one point during the mission everyone opted to return to Konoha due to the conditions of Naruto's injury and that they were mislead about the circumstances of their task. The only one who objected to this and desired to continue assisting this man was Naruto. Okuninushi's fellow travelers derided the hare by offering it harmful advice. The only one who genuinely assisted it was Okuninushi. After this display of magnanimity it predicted that he would be chosen for a husband by the monarch of Inaba, Princess Yakami. Tazuna made a prediction that Naruto's name would be known throughout the world. Both of those predictions have come true.
If it's not inspiration then it's quite the enormous coincidence that an arc in a manga heavily inspired by Far Eastern culture and folklore would happen to resemble numerous characteristics from the aforementioned tale. Especially when you consider that this arc has the only portrayal of rabbits in the entire manga IIRC. A rabbit which happened to stumble upon Naruto's group whom Naruto was the only person that was concerned for its well being.
There's probably still quite a few mythological references that we could make to other characters.