I am reading chapter VI of J. Milne's book, Etale Cohomology. This chapter begins with cohomological dimensions: if $X$ is a finitely generated scheme over a separably closed field $k$, then (étale) cohomological dimension of $X$ is at most $2\dim(X)$. The proof (theorem 1.1 in the book) reduces to the case when $X$ is integral and sheaves of the form $g_{*}\mathcal{F}$ where $g: \eta \longrightarrow X$ is the generic point of $X$ and $\mathcal{F}$ a torsion sheaf on $\eta$.
Lemma 1.2. The sheaf $R^j g_*\mathcal{F}$ has support in dimension $\leq n - j$.
Proof. Let $x \in X$, and $A= \mathcal{O}_{X,x}^{sh}$ (strict Henselization of $\mathcal{O}_{X,x}$), let $A_1,...,A_r$ be quotients of $A$ by minimal prime ideals whose fields of fractions are $K_1,...,K_r$.
First question. Why do we have: $$(R^j g_*\mathcal{F})_{\overline{x}} = \bigoplus H^j(K_r,\mathcal{F}_{\mid K_r})$$ I know that we can compute the LHS as $$(R^j g_*\mathcal{F})_{\overline{x}} = H^j(\eta \times_X \mathrm{Spec}(A), \mathcal{F})$$ so the above may follows once we know that $$\eta \times_X \mathrm{Spec}(A) = \coprod \mathrm{Spec}(K_r)$$ Second question. Suppose that we have such a decomposition, we want to bound the cohomological dimension of those $K_r$, Milne claims that we can do that once the following is true
Lemma 1.3. For any $x \in X$, there exists another $X'/k$ finitely generated and a closed point $x' \in X'$ such that $\mathcal{O}^{sh}_{X,x} \simeq \mathcal{O}^{sh}_{X',x'}$.
I do not see why we need $x'$ to be closed here?