By Properties of Spaces, Lemma \ref{spaces-properties-lemma-dimension-decent-invariant-under-etale} every algebraic space $Y$ \'etale over $X$ has dimension $\leq d$. If $Y$ is quasi-separated, the dimension of $Y$ is equal to the Krull dimension of $|Y|$ by  Decent Spaces, Lemma \ref{decent-spaces-lemma-dimension-decent-space}. Also, if $Y$ is a scheme, then \'etale cohomology of $\mathcal{F}$ over $Y$, resp.\ \'etale cohomology of $\mathcal{F}$ with support in a closed subscheme, agrees with usual cohomology of $\mathcal{F}$, resp.\ usual cohomology with support in the closed subscheme. See Descent, Proposition \ref{descent-proposition-same-cohomology-quasi-coherent} and \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-cohomology-with-support-quasi-coherent}. We will use these facts without further mention. \medskip\noindent By Decent Spaces, Lemma \ref{decent-spaces-lemma-filter-quasi-compact-quasi-separated} there exist an integer $n$ and open subspaces $$ \emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X $$ with the following property: setting $T_p = U_p \setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a quasi-compact separated scheme $V_p$ and a surjective \'etale morphism $f_p : V_p \to U_p$ such that $f_p^{-1}(T_p) \to T_p$ is an isomorphism. \medskip\noindent As $U_n = V_n$ is a scheme, our initial remarks imply the cohomology of $\mathcal{F}$ over $U_n$ vanishes in degrees $> d$ by Cohomology, Proposition \ref{cohomology-proposition-cohomological-dimension-spectral}. Suppose we have shown, by induction, that $H^q(U_{p + 1}, \mathcal{F}|_{U_{p + 1}}) = 0$ for $q > d$. It suffices to show $H_{T_p}^q(U_p, \mathcal{F})$ for $q > d$ is zero in order to conclude the vanishing of cohomology of $\mathcal{F}$ over $U_p$ in degrees $> d$. However, we have $$ H^q_{T_p}(U_p, \mathcal{F}) = H^q_{f_p^{-1}(T_p)}(V_p, \mathcal{F}) $$ by Lemma \ref{lemma-etale-localization-sheaf-with-support} and as $V_p$ is a scheme we obtain the desired vanishing from Cohomology, Proposition \ref{cohomology-proposition-cohomological-dimension-spectral}. In this way we conclude that (1) is true. \medskip\noindent To prove (2) let $U \subset X$ be a quasi-compact open subspace. Consider the open subspace $U' = U \cup U_n$. Let $Z = U' \setminus U$. Then $g : U_n \to U'$ is an \'etale morphism such that $g^{-1}(Z) \to Z$ is an isomorphism. Hence by Lemma \ref{lemma-etale-localization-sheaf-with-support} we have $H^q_Z(U', \mathcal{F}) = H^q_Z(U_n, \mathcal{F})$ which vanishes in degree $> d$ because $U_n$ is a scheme and we can apply Cohomology, Proposition \ref{cohomology-proposition-cohomological-dimension-spectral}. We conclude that $H^d(U', \mathcal{F}) \to H^d(U, \mathcal{F})$ is surjective. Assume, by induction, that we have reduced our problem to the case where $U$ contains $U_{p + 1}$. Then we set $U' = U \cup U_p$, set $Z = U' \setminus U$, and we argue using the morphism $f_p : V_p \to U'$ which is \'etale and has the property that $f_p^{-1}(Z) \to Z$ is an isomorphism. In other words, we again see that $$ H^q_Z(U', \mathcal{F}) = H^q_{f_p^{-1}(Z)}(V_p, \mathcal{F}) $$ and we again see this vanishes in degrees $> d$. We conclude that $H^d(U', \mathcal{F}) \to H^d(U, \mathcal{F})$ is surjective. Eventually we reach the stage where $U_1 = X \subset U$ which finishes the proof. \medskip\noindent A formal argument shows that (2) implies (3).