As I pointed out in my comment, there are too many questions listed here. Maybe I can clarify the term "Jordan-Chevalley decomposition" in the last one. Besides the arXiv post by Danielle Couty and her colleagues which I gave a link to, it's worth looking at the revised published version and its substantial reference list. The article was published in the SMF journal Gazette des Mathematiciens, freely available online here. They look closely at the role Chevalley played, beyond the basic Jordan decomposition of matrices. But it should be emphasized that there are two related frameworks for this kind of decomposition:
1) In the study of Lie algebras over an arbitrary field of characteristic 0, indepndent from the original study of Lie groups over $\mathbb{R}$ or $\mathbb{C}$ by E. Cartan and others, semisimple Lie algebras are of special interest. From the nondegeneracy of the Killing form one sees that the Lie algebra is isomorphic to its derivation algebra. Moreover, in the Jordan decomposition of an adjoint operator, the unique semisimple and nilpotent parts belong to elements of the Lie algbra. Going further, one proves Weyl's complete reducibility theorem and then as a corollary the preservation of the intrinsic Jordan decompsition under any finite dimensional linear representation of the Lie algebra.
Chevalley influenced much of this development, in his books and seminar during the 1950s, but it's hard to document precisely. The Bourbaki group (of which he was an early member) began in 1960 to publish their treatise on Lie groups and Lie algebras. In Chapter I on Lie algebras, there is a crucial proof that complete reducibility in characteristic 0 implies the preservation of Jordan decomposition under homomorphisms: see $\S6$, no. 3, Thm. 3. These results were also given by Serre in his 1965 Harvard lecture notes, with a brief summary in his 1965 Algiers lectures (published by W.A. Benjamin and later translated). In the latter notes, he reverses the order of the key statements, but as far as I can tell one really needs Weyl's theorem first. (This issue comes up in a MO quetiuon here. The proof in section 6 of my 1972 text follows Bourbaki.)
2) But there is a largely separate line of work on linear algebraic groups, which owes even more to Chevalley and certainly merits the label "Jordan-Chevalley decomposition". Actually, a couple of papers by Kolchin in 1948 started in this direction, but Chevalley's 1951 book and his famous 1956-58 classification seminar made the results basic to all further work. The striking fact is that the semisimple and unipotent parts in the multiplicative Jordan decomposition are intrinsically defined in any connected linear algebraic group (over any algebraically closed field, though Chevalley's early work started out over arbitrary fields). Adaptations to fields of definition then follow.
It is this multiplicative version that Couty et al. emphasize in their work, which is partly historical and partly pedagogical.
