The three questions asked are fairly elementary, as the comment by LSpice indicates; in the format here, it's best to avoid multiple questions however. Aside from this, it's probably more natural to think of choosing canonical representatives for a Weyl group in purely algebraic terms. For this a good reference would be the lecture notes written up at Yale in 1967-68 as Robert Steinberg gave his course there. Though formerly quite readily accessible online (for instance in the original typewritten format), it's harder to locate a PDF version now that there is a somewhat edited AMS softcover publication using LaTeX: Lectures on Chevalley Groups, University Lecture Series, Vol. 66 (2016).
Here the exponentials are formal when the field has prime characteristic, but the ideas are the same as in characteristic 0. For the split groups, see especially Chapter 3. For non-split but quasi-split groups see Chapter 11; but here the "Weyl group" is a subgroup of the usual one, so extra care must be taken to include this situation. In any case, the canonical representatives can be chosen essentially as indicated: fix an arbitrary reduced expression and take a product of expressions for the various simple roots involved. (In the quasi-split case, the procedure is similar.)
By the way, it's more precise to use tags here such as 'weyl-groups', 'reference-request', 'algebraic-groups', along with 'lie-algebras'.
ADDED: Concerning the term "canonical" used here, it's potentially rather delicate when choosing representatives of Weyl group elements. Typically the choices (say in a split group) differ by an element of a maximal torus (Cartan subgroup). Indeed, the Weyl group itself may or many not embed naturally into the given group. For most practical purposes it's often enough to fix various data involved and then work with the resulting coset representatives.