Trivial observations: (6) is a special case of (5) and (4) is a special case of (3). We first prove that cases (5) and (3) reduce to case (1). Namely, by bootstrapping the diagonal Lemma \ref{lemma-bootstrap-diagonal} we see that (3) implies (5). In case (5) we set $R = U \times_F U$ which is an algebraic space by assumption. Moreover, by assumption both projections $s, t : R \to U$ are surjective, flat and locally of finite presentation. The map $j : R \to U \times_S U$ is clearly an equivalence relation. By Lemma \ref{lemma-surjective-flat-locally-finite-presentation} the map $U \to F$ is a surjection of sheaves. Thus $F = U/R$ which reduces us to case (1). \medskip\noindent Next, we show that (1) reduces to (2). Namely, let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \to U \times_S U$ is an equivalence relation. Choose a scheme $U'$ and a surjective \'etale morphism $U' \to U$. Let $R' = R|_{U'}$ be the restriction of $R$ to $U'$. By Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-quotient-pre-equivalence-relation-restrict} we see that $U/R = U'/R'$. Since $s', t' : R' \to U'$ are also flat and locally of finite presentation (see More on Groupoids in Spaces, Lemma \ref{spaces-more-groupoids-lemma-restrict-preserves-type}) this reduces us to the case where $U$ is a scheme. As $j$ is an equivalence relation we see that $j$ is a monomorphism. As $s : R \to U$ is locally of finite presentation we see that $j : R \to U \times_S U$ is locally of finite type, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-permanence-finite-type}. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite} we see that $j$ is locally quasi-finite and separated. Hence if $U$ is a scheme, then $R$ is a scheme by Morphisms of Spaces, Proposition \ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}. Thus we reduce to proving the theorem in case (2). \medskip\noindent Assume $F = U/R$ where $(U, R, s, t, c)$ is a groupoid scheme over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \to U \times_S U$ is an equivalence relation. By Lemma \ref{lemma-slice-equivalence-relation} we reduce to that case where $s, t$ are flat, locally of finite presentation, and locally quasi-finite. Let $U = \bigcup_{i \in I} U_i$ be an affine open covering (with index set $I$ of cardinality $\leq$ than the size of $U$ to avoid set theoretic problems later -- most readers can safely ignore this remark). Let $(U_i, R_i, s_i, t_i, c_i)$ be the restriction of $R$ to $U_i$. It is clear that $s_i, t_i$ are still flat, locally of finite presentation, and locally quasi-finite as $R_i$ is the open subscheme $s^{-1}(U_i) \cap t^{-1}(U_i)$ of $R$ and $s_i, t_i$ are the restrictions of $s, t$ to this open. By Lemma \ref{lemma-better-finding-opens} (or the simpler Spaces, Lemma \ref{spaces-lemma-finding-opens}) the map $U_i/R_i \to U/R$ is representable by open immersions. Hence if we can show that $F_i = U_i/R_i$ is an algebraic space, then $\coprod_{i \in I} F_i$ is an algebraic space by Spaces, Lemma \ref{spaces-lemma-coproduct-algebraic-spaces}. As $U = \bigcup U_i$ is an open covering it is clear that $\coprod F_i \to F$ is surjective. Thus it follows that $U/R$ is an algebraic space, by Spaces, Lemma \ref{spaces-lemma-glueing-algebraic-spaces}. In this way we reduce to the case where $U$ is affine and $s, t$ are flat, locally of finite presentation, and locally quasi-finite and $j$ is an equivalence. \medskip\noindent Assume $(U, R, s, t, c)$ is a groupoid scheme over $S$, with $U$ affine, such that $s, t$ are flat, locally of finite presentation, and locally quasi-finite, and $j$ is an equivalence relation. Choose $u \in U$. We apply More on Groupoids in Spaces, Lemma \ref{spaces-more-groupoids-lemma-quasi-splitting-affine-scheme} to $u \in U, R, s, t, c$. We obtain an affine scheme $U'$, an \'etale morphism $g : U' \to U$, a point $u' \in U'$ with $\kappa(u) = \kappa(u')$ such that the restriction $R' = R|_{U'}$ is quasi-split over $u'$. Note that the image $g(U')$ is open as $g$ is \'etale and contains $u'$. Hence, repeatedly applying the lemma, we can find finitely many points $u_i \in U$, $i = 1, \ldots, n$, affine schemes $U'_i$, \'etale morphisms $g_i : U_i' \to U$, points $u'_i \in U'_i$ with $g(u'_i) = u_i$ such that (a) each restriction $R'_i$ is quasi-split over some point in $U'_i$ and (b) $U = \bigcup_{i = 1, \ldots, n} g_i(U'_i)$. Now we rerun the last part of the argument in the preceding paragraph: Using Lemma \ref{lemma-better-finding-opens} (or the simpler Spaces, Lemma \ref{spaces-lemma-finding-opens}) the map $U'_i/R'_i \to U/R$ is representable by open immersions. If we can show that $F_i = U'_i/R'_i$ is an algebraic space, then $\coprod_{i \in I} F_i$ is an algebraic space by Spaces, Lemma \ref{spaces-lemma-coproduct-algebraic-spaces}. As $\{g_i : U'_i \to U\}$ is an \'etale covering it is clear that $\coprod F_i \to F$ is surjective. Thus it follows that $U/R$ is an algebraic space, by Spaces, Lemma \ref{spaces-lemma-glueing-algebraic-spaces}. In this way we reduce to the case where $U$ is affine and $s, t$ are flat, locally of finite presentation, and locally quasi-finite, $j$ is an equivalence, and $R$ is quasi-split over $u$ for some $u \in U$. \medskip\noindent Assume $(U, R, s, t, c)$ is a groupoid scheme over $S$, with $U$ affine, $u \in U$ such that $s, t$ are flat, locally of finite presentation, and locally quasi-finite and $j = (t, s) : R \to U \times_S U$ is an equivalence relation and $R$ is quasi-split over $u$. Let $P \subset R$ be a quasi-splitting of $R$ over $u$. By Lemma \ref{lemma-divide-subgroupoid} we see that $(U, R, s, t, c)$ is the restriction of a groupoid $(\overline{U}, \overline{R}, \overline{s}, \overline{t}, \overline{c})$ by a surjective finite locally free morphism $U \to \overline{U}$ such that $P = U \times_{\overline{U}} U$. Note that $s, t$ are the base changes of the morphisms $\overline{s}, \overline{t}$ by $U \to \overline{U}$. As $\{U \to \overline{U}\}$ is an fppf covering we conclude $\overline{s}, \overline{t}$ are flat, locally of finite presentation, and locally quasi-finite, see Descent, Lemmas \ref{descent-lemma-descending-property-flat}, \ref{descent-lemma-descending-property-locally-finite-presentation}, and \ref{descent-lemma-descending-property-quasi-finite}. Consider the commutative diagram $$ \xymatrix{ U \times_{\overline{U}} U \ar@{=}[r] \ar[rd] & P \ar[r] \ar[d] & R \ar[d] \\ & \overline{U} \ar[r]^{\overline{e}} & \overline{R} } $$ It is a general fact about restrictions that the outer four corners form a cartesian diagram. By the equality we see the inner square is cartesian. Since $P$ is open in $R$ (by definition of a quasi-splitting) we conclude that $\overline{e}$ is an open immersion by Descent, Lemma \ref{descent-lemma-descending-property-open-immersion}. An application of Groupoids, Lemma \ref{groupoids-lemma-quotient-pre-equivalence-relation-restrict} shows that $U/R = \overline{U}/\overline{R}$. Hence we have reduced to the case where $(U, R, s, t, c)$ is a groupoid scheme over $S$, with $U$ affine, $u \in U$ such that $s, t$ are flat, locally of finite presentation, and locally quasi-finite and $j = (t, s) : R \to U \times_S U$ is an equivalence relation and $e : U \to R$ is an open immersion! \medskip\noindent But of course, if $e$ is an open immersion and $s, t$ are flat and locally of finite presentation then the morphisms $t, s$ are \'etale. For example you can see this by applying More on Groupoids, Lemma \ref{more-groupoids-lemma-sheaf-differentials} which shows that $\Omega_{R/U} = 0$ which in turn implies that $s, t : R \to U$ is G-unramified (see Morphisms, Lemma \ref{morphisms-lemma-unramified-omega-zero}), which in turn implies that $s, t$ are \'etale (see Morphisms, Lemma \ref{morphisms-lemma-flat-unramified-etale}). And if $s, t$ are \'etale then finally $U/R$ is an algebraic space by Spaces, Theorem \ref{spaces-theorem-presentation}.