Recall that (3) is the definition of (2), see Algebraic Stacks, Definition \ref{algebraic-definition-deligne-mumford}. The implication (3) $\Rightarrow$ (1) is Lemma \ref{lemma-properties-covering-imply-diagonal}. Assume (1). Let $x \in |\mathcal{X}|$ be a finite type point. We will produce a scheme over $\mathcal{X}$ which ``works'' in a neighbourhood of $x$. At the end of the proof we will take the disjoint union of all of these to conclude. \medskip\noindent By Lemma \ref{lemma-point-finite-type-monomorphism} the residual gerbe $\mathcal{Z}_x$ of $\mathcal{X}$ at $x$ exists and $\mathcal{Z}_x \to \mathcal{X}$ is locally of finite type. By Lemma \ref{lemma-separation-properties-residual-gerbe} the algebraic stack $\mathcal{Z}_x$ is DM. By Lemma \ref{lemma-DM-residual-gerbe} there exists a field $k$ and a surjective \'etale morphism $z : \Spec(k) \to \mathcal{Z}_x$. In particular the composition $x : \Spec(k) \to \mathcal{X}$ is locally of finite type (by Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-composition-finite-type} and \ref{spaces-morphisms-lemma-etale-locally-finite-type}). \medskip\noindent Pick a scheme $U$ and a smooth morphism $U \to \mathcal{X}$ such that $x$ is in the image of $|U| \to |\mathcal{X}|$. Consider the following fibre square $$ \xymatrix{ U \ar[d] & F  \ar[l] \ar[d] \\ \mathcal{X} & \Spec(k) \ar[l]_-x } $$ in other words $F = U \times_{\mathcal{X}, x} \Spec(k)$. By Properties of Stacks, Lemma \ref{stacks-properties-lemma-points-cartesian} we see that $F$ is nonempty. As $\mathcal{Z}_x \to \mathcal{X}$ is a monomorphism we have $$ \Spec(k) \times_{z, \mathcal{Z}_x, z} \Spec(k) = \Spec(k) \times_{x, \mathcal{X}, x} \Spec(k) $$ with \'etale projection maps to $\Spec(k)$ by construction of $z$. Since $$ F \times_U F = (\Spec(k) \times_\mathcal{X} \Spec(k)) \times_{\Spec(k)} F $$ we see that the projections maps $F \times_U F \to F$ are \'etale as well. It follows that $\Delta_{F/U} : F \to F \times_U F$ is \'etale (see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-permanence}). By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-universally-injective-open} this implies that $\Delta_{F/U}$ is an open immersion, which finally implies by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-diagonal-unramified-morphism} that $F \to U$ is unramified. \medskip\noindent Pick a nonempty affine scheme $V$ and an \'etale morphism $V \to F$. (This could be avoided by working directly with $F$, but it seems easier to explain what's going on by doing so.) Picture $$ \xymatrix{ U \ar[d] & F  \ar[l] \ar[d] & V \ar[l] \ar[ld] \\ \mathcal{X} & \Spec(k) \ar[l]_-x } $$ Then $V \to \Spec(k)$ is a smooth morphism of schemes and $V \to U$ is an unramified morphism of schemes (see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-composition-smooth} and \ref{spaces-morphisms-lemma-composition-unramified}). Pick a closed point $v \in V$ with $k \subset \kappa(v)$ finite separable, see Varieties, Lemma \ref{varieties-lemma-smooth-separable-closed-points-dense}. Let $u \in U$ be the image point. The local ring $\mathcal{O}_{V, v}$ is regular (see Varieties, Lemma \ref{varieties-lemma-smooth-regular}) and the local ring homomorphism $$ \varphi : \mathcal{O}_{U, u} \longrightarrow \mathcal{O}_{V, v} $$ coming from the morphism $V \to U$ is such that $\varphi(\mathfrak m_u)\mathcal{O}_{V, v} = \mathfrak m_v$, see Morphisms, Lemma \ref{morphisms-lemma-unramified-at-point}. Hence we can find $f_1, \ldots, f_d \in \mathcal{O}_{U, u}$ such that the images $\varphi(f_1), \ldots, \varphi(f_d)$ form a basis for $\mathfrak m_v/\mathfrak m_v^2$ over $\kappa(v)$. Since $\mathcal{O}_{V, v}$ is a regular local ring this implies that $\varphi(f_1), \ldots, \varphi(f_d)$ form a regular sequence in $\mathcal{O}_{V, v}$ (see Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}). After replacing $U$ by an open neighbourhood of $u$ we may assume $f_1, \ldots, f_d \in \Gamma(U, \mathcal{O}_U)$. After replacing $U$ by a possibly even smaller open neighbourhood of $u$ we may assume that $V(f_1, \ldots, f_d) \to \mathcal{X}$ is flat and locally of finite presentation, see Lemma \ref{lemma-slice}. By construction $$ V(f_1, \ldots, f_d) \times_\mathcal{X} \Spec(k) \longleftarrow V(f_1, \ldots, f_d) \times_\mathcal{X} V $$ is \'etale and $V(f_1, \ldots, f_d) \times_\mathcal{X} V$ is the closed subscheme $T \subset V$ cut out by $f_1|_V, \ldots, f_d|_V$. Hence by construction $v \in T$ and $$ \mathcal{O}_{T, v} = \mathcal{O}_{V, v}/(\varphi(f_1), \ldots, \varphi(f_d)) = \kappa(v) $$ a finite separable extension of $k$. It follows that $T \to \Spec(k)$ is unramified at $v$, see Morphisms, Lemma \ref{morphisms-lemma-unramified-at-point}. By definition of an unramified morphism of algebraic spaces this means that $V(f_1, \ldots, f_d) \times_\mathcal{X} \Spec(k) \to \Spec(k)$ is unramified at the image of $v$ in $V(f_1, \ldots, f_d) \times_\mathcal{X} \Spec(k)$. Applying Lemma \ref{lemma-etale-at-point} we see that on shrinking $U$ to yet another open neighbourhood of $u$ the morphism $V(f_1, \ldots, f_d) \to \mathcal{X}$ is \'etale. \medskip\noindent We conclude that for every finite type point $x$ of $\mathcal{X}$ there exists an \'etale morphism $f_x : W_x \to \mathcal{X}$ with $x$ in the image of $|f_x|$. Set $W = \coprod_x W_x$ and $f = \coprod f_x$. Then $f$ is \'etale. In particular the image of $|f|$ is open, see Properties of Stacks, Lemma \ref{stacks-properties-lemma-topology-points}. By construction the image contains all finite type points of $\mathcal{X}$, hence $f$ is surjective by Lemma \ref{lemma-enough-finite-type-points} (and Properties of Stacks, Lemma \ref{stacks-properties-lemma-characterize-surjective}).