Cases (2) and (3) follow immediately from case (1) but we will give a separate proofs of (2) and (3) as these proofs use significantly less theory. \medskip\noindent Proof of (3). Let $U$ be an affine scheme and let $U \to X$ be an \'etale morphism. Set $R = U \times_X U$. The two projection morphisms $s, t : R \to U$ are \'etale morphisms of schemes. By Properties of Spaces, Definition \ref{spaces-properties-definition-dimension} we see that $\dim(U) = 0$ and $\dim(R) = 0$. Since $R$ is a locally Noetherian scheme of dimension $0$, we see that $R$ is a disjoint union of spectra of Artinian local rings (Properties, Lemma \ref{properties-lemma-locally-Noetherian-dimension-0}). Since we assumed that $X$ is Noetherian (so quasi-separated) we conclude that $R$ is quasi-compact. Hence $R$ is an affine scheme (use Schemes, Lemma \ref{schemes-lemma-disjoint-union-affines}). The \'etale morphisms $s, t : R \to U$ induce finite residue field extensions. Hence $s$ and $t$ are finite by Algebra, Lemma \ref{algebra-lemma-essentially-of-finite-type-into-artinian-local} (small detail omitted).  Thus Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence} shows that $X = U/R$ is an affine scheme. \medskip\noindent Proof of (2) -- almost identical to the proof of (4). Let $U$ be an affine scheme and let $U \to X$ be an \'etale morphism. Set $R = U \times_X U$. The two projection morphisms $s, t : R \to U$ are \'etale morphisms of schemes. By Properties of Spaces, Definition \ref{spaces-properties-definition-dimension} we see that $\dim(U) = 0$ and similarly $\dim(R) = 0$. On the other hand, the morphism $U \to \Spec(k)$ is locally of finite type as the composition of the \'etale morphism $U \to X$ and $X \to \Spec(k)$, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-composition-finite-type} and \ref{spaces-morphisms-lemma-etale-locally-finite-type}. Similarly, $R \to \Spec(k)$ is locally of finite type. Hence by Varieties, Lemma \ref{varieties-lemma-algebraic-scheme-dim-0} we see that $U$ and $R$ are disjoint unions of spectra of local Artinian $k$-algebras finite over $k$. The same thing is therefore true of $U \times_{\Spec(k)} U$. As $$ R = U \times_X U \longrightarrow U \times_{\Spec(k)} U $$ is a monomorphism, we see that $R$ is a finite(!) union of spectra of finite $k$-algebras. It follows that $R$ is affine, see Schemes, Lemma \ref{schemes-lemma-disjoint-union-affines}. Applying Varieties, Lemma \ref{varieties-lemma-algebraic-scheme-dim-0} once more we see that $R$ is finite over $k$. Hence $s, t$ are finite, see Morphisms, Lemma \ref{morphisms-lemma-finite-permanence}. Thus Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence} shows that the open subspace $U/R$ of $X$ is an affine scheme. Since the schematic locus of $X$ is an open subspace (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme}), and since $U \to X$ was an arbitrary \'etale morphism from an affine scheme we conclude that $X$ is a scheme. \medskip\noindent Proof of (1). By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-vanishing-above-dimension} we have vanishing of higher cohomology groups for all quasi-coherent sheaves $\mathcal{F}$ on $X$. Hence $X$ is affine (in particular a scheme) by Cohomology of Spaces, Proposition \ref{spaces-cohomology-proposition-vanishing-affine}.