I got that much, or rather that having both Sharingan and Mokuton were needed to use Izanagi.
Eh, not really. There would be no point in the creation of Izanami if only those with both Senju and Uchiha cells could use Izanagi. It's just that Izanagi is an inferior version of a Sage of the Six Paths technique (read, Rinnegan technique) that was used to create the Bijuu from the Juubi. Having the "other half" through integration Hashirama's cells let's you boost the technique (in the case of Obito) or sustain using it (in the case of Danzo).
But as far as I know, none of the Senju except Hashirama had Mokuton, so I don't think it's fair to call it their bloodline. Rather than "body and eyes" think "body and mind." Of the Sage's two children, one of inherited his subtle and cunning mind, and one inherited his strength, of character as well as body. Of the eldest's (mind) line we have the Uchiha. Of the younger's (body) issue we have the Senju, the First, Second and Fifth Hokages, and the Uzumaki, a group that apparently includes not only Kushina and Naruto, but Nagato and Karin. All the Uzumaki were supposedly really long lived, because of their strong and vital chakra, Hashirama and Tsunade are the greatest healers the world has ever seen, Kushina held on for a bit after Kyuubi was extracted where all the other jinchuurki died immediately. All of these I can easily attribute to having strong chakra with a decidedly physical bent to it.
As to the issue of Mokuton, no, I'm sticking with that as the kekkei genkai. Keep in mind that Amaterasu is Enton, a kekkei genkai that combines Fire and Lightning elemental chakra just like Mokuton combines Earth and Water. One is unique to Uchiha, and the other is unique to Senju. Just because Hashirama is the only one Senju to manage to unlock that ability doesn't mean it isn't their kekkei genkai. Not all Uchiha can unlock the Sharingan, let alone the Mangekyou Sharingan, which is necessary in order to unlock Enton. However, we do know that this is a significant difference between the Uchiha and Senju. The Uchiha were naturally aggressive and it has been stated they devoted much time and effort into learning how to unlock the Sharingan and advance it further into the Mangekyou Sharingan, and that this knowledge was shared throughout the clan. Even then, not everyone was capable of it. Contrast with the Senju who were also strong ninja, but they sought peace instead of domination and its likely that after Hashirama developed Mokuton, others simply let it be when those that tried to duplicate his feat were unable to do so, rather than investing time like the Uchiha.
For your Body vs Mind, I'd argue that the Senju got the conniving mind given the way they made peace with the Uchiha and then managed to utterly marginalize such a powerful clan. Other than that's its stated that the Uchiha ancestor got the Eyes and Spiritual energy (Yin chakra, with genjutsu happening to be included under techniques that require Yin chakra), while the Senju ancestor got his powerful body and physical energy (Yang chakra, which broadly covers enhancing/manipulating living things, like say Mokuton or healing display).
Don't forget he had all of Nagato's powers, with about a thousand times more chakra to back them up, thanks to the Juubi. Remember that time Nagato resurrected all of Konoha at the expense of his life? I imagine that might be mildly strenuous to the Sage.
Again, I like to think that he was sort of like Naruto, and gained his power through sheer determination and force of character.
Oh, I didn't. I just thought pointing out that he had the Rinnegan, when that's one of his defining characteristics was unnecessary, but you're right. Nagato himself pointed out how pitiful he is in comparison the the Sage when he used the Moon creation technique on 8-tails Naruto and not only failed to seal him, failed to produce something anywhere close to approaching the size of the Moon. I also wonder about how powerful the Sage was without the Juubi. When he created the Bijuu and sealed the Juubi's body in the Moon, I would imagine he was no longer able to draw upon the power of the Juubi to perform those feats in the first place.
The Third is called the “God of Ninja” because of his power, but more so because of his skill. He is the single most knowledgeable ninja in the world. Keep in mind his other epithet, “The Professor.” Sarutobi Hiruzen knew every single jutsu (including the Forbidden ones) that existed within Konoha (that he was capable of learning, obviously he can’t learn other people’s bloodline skills) and was considered a master of every form of combat. The guy was basically the original Kakashi (the "Copy Ninja”), but through sheer hard work and skill since he didn’t have a Sharingan providing him with cheat codes to make the process go faster. Also, it would seem that he is the one who started the trend of having personal summons.
None of this is inconsistent with his being an expy/shout out of Sun Wukong. Read the link.
My comment is mostly about people saying, "Oh, 'God of Ninja,' then why wasn't he rearranging land masses like Hashirama and Madara?" I didn't mean to take anything away from you saying he is like Sun Wukong.