To help answer Question 1, Milnor proved a local-global theorem for Witt rings of global fields. Recall that The Grothendieck-Witt ring $\widehat{W}(k)$ of a field $k$ is the ring obtained by starting with the free abelian group on isomorphism classes of quadratic modules and moding out by the ideal generated by symbols of the form $[M]+[N]-[M']-[N']$, whenever $[M]\oplus[N]\simeq [M']+[N']$. The multiplication comes from tensor product of quadratic modules. There is a special quadratic module $H$ given by $x^2-y^2=0$. This is the hyperbolic module. The Witt ring $W(k)$ of a field $k$ is the quotient of $\widehat{W}(k)$ by the ideal generated by $[H]$.

Now, the main theorem of Milnor's paper is that there is a split exact sequence $$0\rightarrow W(k)\rightarrow W(k(t))\rightarrow \oplus\_\pi W(\overline{k(t)}\_\pi)\rightarrow 0,$$ where $\pi$ runs over all irreducible monic polynomials in $k[t]$, and $\overline{k(t)}\_\pi$ denotes the residue field of the completion of $k(t)$ at $\pi$.

The morphisms $W(k(t))\rightarrow W(\overline{k(t)}\_\pi)$ come from first the map $W(k(t))\rightarrow W(k(t)\_\pi)$. Then, there is a map $W(k(t)\_\pi)\rightarrow W(\overline(k(t))\_\pi)$ that sends the quadratic module $u\pi x^2=0$ to $ux^2=0$, where $u$ is any unit of the local field.

Interestingly, Milnor $K$-theory is not used in the proof. However, the proof for Witt rings closely models the proof of a similar fact for Milnor $K$-theory: the sequence $$0\rightarrow K_n^M(k)\rightarrow K_n^M(k(t))\rightarrow\oplus\_\pi K_{n-1}^M(\overline{k(t)}\_\pi)\rightarrow 0.$$

The important new perspective is the formal symbolic perspective, which was already existent for lower $K$-groups, but is very fruitful for studying the Witt ring as well.