You are asking several questions here, so it may be useful to separate out what is going on first in the setting of affine reflection groups. This is independent of the application to algebraic groups or loop groups.

The answer to your first question (how to write $s_0$ in terms of the presentation) is straightforward, though different sources tend to use different notations and terminology. In the set-up of my textbook referenced below, the canonical generators of an affine Weyl group are given by the usual simple reflections $s_1, \dots, s_n$ in the underlying finite Weyl group together with one other reflection $s_0$ defined to be $s_{\tilde{\alpha},1}$. Here $\tilde{\alpha}$ is the *highest* root in the finite root system. The affine reflection $s_0$ acts on a typical $\lambda$ by the explicit formula
$s_0 \lambda := \lambda - (\langle \lambda, \tilde{\alpha} \rangle - 1) \tilde{\alpha}^\vee$ (where the last symbol means the coroot of the highest root). It gets messy here to juggle the roots and weights along with coroots and coweights, but the geometric idea is fairly transparent when you draw a sketch.

Historically, the affine Weyl groups related to Chevalley groups over $p$-adic fields were probably first studied in an influential 1965 paper by Iwahori and Matsumoto here.
Since affine Weyl groups arise also in other contexts (compact Lie groups, loop groups, characteristic $p$ algebraic groups), Bourbaki gave a more general and formal treatment in Chapters IV-VI of *Groupes et algebres de Lie* (1968). There is a somewhat differently organized treatment in my *Reflection Groups and Coxeter Groups* (Cambridge, 1990), Chapter 4.

Note that the characteristic $p$ version (first articulated by Verma) works with a version of affine Weyl groups corresponding to the *Langlands dual* situation, which is another complication in dealing with the notation.

Concerning your more specific question about loop groups, keep in mind that the affine Weyl group (like the finite Weyl group) typically doesn't *embed* into the larger group. In any case, the basic approach to working with the internal structure of the group over a field of formal power series goes back to the paper of Iwahori-Matsumoto on $p$-adic groups.