For any morphism $V \to Y$ the complex $K_V$ is pseudo-coherent (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pseudo-coherent-pullback}) and $E_V$ is $V$-perfect (More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-base-change-relatively-perfect}). Another observation is that given $y : \Spec(k) \to Y$ and a field extension $k'/k$ with $y' : \Spec(k') \to Y$ the induced morphism, we have $$ \Ext^i_{\mathcal{O}_{X_{y'}}}(K_{y'}, E_{y'}) = \Ext^i_{\mathcal{O}_{X_y}}(K_y, E_y) \otimes_k k' $$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-morphism-and-hom-out-of-perfect}. Thus the vanishing in (1) is really a property of the induced point $y \in |Y|$. We will use these two observations without further mention in the proof. \medskip\noindent Assume first $Y$ is an affine scheme. Then we may apply More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-compute-ext-rel-perfect} and find a pseudo-coherent $L \in D(\mathcal{O}_Y)$ which ``universally computes'' $Rf_*R\SheafHom(K, E)$ in the sense described in that lemma. Unwinding the definitions, we obtain for a point $y \in Y$ the equality $$ \Ext^i_{\kappa(y)}(L \otimes_{\mathcal{O}_Y}^\mathbf{L} \kappa(y), \kappa(y)) = \Ext^i_{\mathcal{O}_{X_y}}(K_y, E_y) $$ We conclude that $$ H^i(L \otimes_{\mathcal{O}_Y}^\mathbf{L} \kappa(y)) = 0 \text{ for } i > 0 \Leftrightarrow \Ext^i_{\mathcal{O}_{X_y}}(K_y, E_y) = 0 \text{ for }i < 0. $$ By Derived Categories of Schemes, Lemma \ref{perfect-lemma-jump-loci} the set $W$ of $y \in Y$ where this happens defines an open of $Y$. This open $W$ then satisfies the requirement in (1) for all morphisms from spectra of fields, by the ``universality'' of $L$. \medskip\noindent Let's go back to $Y$ a general algebraic space. Choose an \'etale covering $\{V_i \to Y\}$ by affine schemes $V_i$. Then we see that the subset $W \subset |Y|$ pulls back to the corresponding subset $W_i \subset |V_i|$ for $X_{V_i}$, $K_{V_i}$, $E_{V_i}$. By the previous paragraph we find that $W_i$ is open, hence $W$ is open. This proves (1) in general. Moreover, parts (2) and (3) are entirely formulated in terms of the category $\textit{Spaces}/W$ and the restrictions $X_W$, $K_W$, $E_W$. This reduces us to the case $W = Y$. \medskip\noindent Assume $W = Y$. We claim that for any algebraic space $V$ over $Y$ we have $Rf_{V, *}R\SheafHom(K_V, E_V)$ has vanishing cohomology sheaves in degrees $< 0$. This will prove (2) because $$ \Ext^i_{\mathcal{O}_{X_V}}(K_V, E_V) = H^i(X_V, R\SheafHom(K_V, E_V)) =  H^i(V, Rf_{V, *}R\SheafHom(K_V, E_V)) $$ by Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-section-RHom-over-U} and \ref{sites-cohomology-lemma-Leray-unbounded} and the vanishing of the cohomology sheaves implies the cohomology group $H^i$ is zero for $i < 0$ by Derived Categories, Lemma \ref{derived-lemma-negative-vanishing}. \medskip\noindent To prove the claim, we may work \'etale locally on $V$. In particular, we may assume $Y$ is affine and $W = Y$. Let $L \in D(\mathcal{O}_Y)$ be as in the second paragraph of the proof. For an algebraic space $V$ over $Y$ denote $L_V$ the derived pullback of $L$ to $V$. (An important feature we will use is that $L$ ``works'' for all algebraic spaces $V$ over $Y$ and not just affine $V$.) As $W = Y$ we have $H^i(L) = 0$ for $i > 0$ (use More on Algebra, Lemma \ref{more-algebra-lemma-lift-pseudo-coherent-from-residue-field} to go from fibres to stalks). Hence $H^i(L_V) = 0$ for $i > 0$. The property defining $L$ is that $$ Rf_{V, *}R\SheafHom(K_V, E_V) = R\SheafHom(L_V, \mathcal{O}_V) $$ Since $L_V$ sits in degrees $\leq 0$, we conclude that $R\SheafHom(L_V, \mathcal{O}_V)$ sits in degrees $\geq 0$ thereby proving the claim. This finishes the proof of (2). \medskip\noindent Assume $W = Y$ but make no assumptions on the algebraic space $Y$. Since we have (2), we see from Simplicial Spaces, Lemma \ref{spaces-simplicial-lemma-fppf-neg-ext-zero-hom} that the functor $F$ given by $F(V) = \Hom_{\mathcal{O}_{X_V}}(K_V, E_V)$ is a sheaf\footnote{To check the sheaf property for a covering $\{V_i \to V\}_{i \in I}$ first consider the {\v C}ech fppf hypercovering $a : V_\bullet \to V$ with $V_n = \coprod_{i_0 \ldots i_n} V_{i_0} \times_V \ldots \times_V V_{i_n}$ and then set $U_\bullet = V_\bullet \times_{a, V} X_V$. Then $U_\bullet \to X_V$ is an fppf hypercovering to which we may apply Simplicial Spaces, Lemma \ref{spaces-simplicial-lemma-fppf-neg-ext-zero-hom}.} on $(\textit{Spaces}/Y)_{fppf}$. Thus to prove that $F$ is an algebraic space and that $F \to Y$ is affine and of finite presentation, we may work \'etale locally on $Y$; see Bootstrap, Lemma \ref{bootstrap-lemma-locally-algebraic-space-finite-type} and Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-affine-local} and \ref{spaces-morphisms-lemma-finite-presentation-local}. We conclude that it suffices to prove $F$ is an affine algebraic space of finite presentation over $Y$ when $Y$ is an affine scheme. In this case we go back to our pseudo-coherent complex $L \in D(\mathcal{O}_Y)$. Since $H^i(L) = 0$ for $i > 0$, we can represent $L$ by a complex of the form $$ \ldots \to \mathcal{O}_Y^{\oplus m_1} \to \mathcal{O}_Y^{\oplus m_0} \to 0 \to \ldots $$ with the last term in degree $0$, see More on Algebra, Lemma \ref{more-algebra-lemma-pseudo-coherent}. Combining the two displayed formulas earlier in the proof we find that $$ F(V) = \Ker( \Hom_V(\mathcal{O}_V^{\oplus m_0}, \mathcal{O}_V) \to  \Hom_V(\mathcal{O}_V^{\oplus m_1}, \mathcal{O}_V) ) $$ In other words, there is a fibre product diagram $$ \xymatrix{ F \ar[d] \ar[r] & Y \ar[d]^0 \\ \mathbf{A}_Y^{m_0} \ar[r] & \mathbf{A}_Y^{m_1} } $$ which proves what we want.