Since Joel has already answered Questions (1), and (2), I will only offer an answer for Question 3.

This is a revised version of my answer; thanks to Joel Hamkins for pointing out that my previous construction was not quite right.

Start with a with a simple graph with 3 elements {$a,b,c$}, where each of the three nodes has an edge to the other two. So in this 3-element model of "set theory", $a$ = {$b$, $c$}, $b$ = {$c$, $a$}, and $c$ = {$a$, $b$}.

Given a digraph $G=(X,E)$, with $X$ as the vertex set and $E$ as the edge set, define the deficiency set $D(G)$ of $G$ to be the collection of subsets $S$ of $X$ that are not "coded" in $G$, i.e., there is no element $a$ in $X$ such that $S$ = {$x \in X : xEa$}.

We now can define by recursion a digraph $G_\alpha = (X_\alpha, E_\alpha)$ for each ordinal $\alpha$ as follows:

$G_0 = G$;

$G_{\alpha+1} = (X_{\alpha+1}, E_{\alpha+1})$, where $X_{\alpha+1} = X_{\alpha} \cup D(G_{\alpha})$, and $E_{\alpha+1} = E_{\alpha}$ together with edges of the form $(x,X)$, where $x\in X_{\alpha}$, $X \in D(G_{\alpha})$, and $x\in X$.

For limit $\alpha$, $G_\alpha$ is the union of $G_\beta$ for $\beta<\alpha$.

The model/digraph we are interested in is the union of all the $G_\alpha$, as $\alpha$ ranges over the ordinals.

This model, let's call it $V(a,b,c)$, satisfies all the axioms of $ZF$ with the exception of Foundation.

$V(a,b,c)$ also satisfies $S$ since any infinite descending epsilon chain must eventually hit $a$, $b$, or $c$.

So this shows that Question 3 has a negative answer.

Since {$a,b,c$} are indiscernibles in $V(a,b,c)$, $DC$ fails in $V(a,b,c)$, but a variation on this theme might produce a model with enough asymmetry for $DC$ to hold as well.

PS. Models of $ZF$ in which the Foundation fails are often constructed using the so-called Bernays-Rieger permutation method (not to be confused with the Fraenkel-Mostowski permutation method of constructing models of $ZF$ in which the axiom of choice fails). The model constructed above is based on a different idea, explored in detail for models of finite set theory in the following paper:

A. Enayat, J. Schmerl, and A. Visser, Omega Models of Finite Set Theory , to appear in Set theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies (edited by J. Kennedy and R. Kossak), Cambridge University Press, to appear October 2011.

A preprint can be found here.

Since Joel has already answered Questions (1), and (2), I will only offer an answer for Question 3.

This is a revised version of my answer; thanks to Joel Hamkins for pointing out that my previous construction was not quite right.

Start with a simple graph with 3 elements {$a,b,c$}, where each of the three nodes has an edge to the other two. So in this 3-element model of "set theory", $a$ = {$b$, $c$}, $b$ = {$c$, $a$}, and $c$ = {$a$, $b$}.

Given an extensional digraph $G=(X,E)$, with $X$ as the vertex set and $E$ as the edge set, define the deficiency set $D(G)$ of $G$ to be the collection of subsets $S$ of $X$ that are not "coded" in $G$, i.e., there is no element $a$ in $X$ such that $S$ = {$x \in X : xEa$}.

We now can define by recursion a digraph $G_\alpha = (X_\alpha, E_\alpha)$ for each ordinal $\alpha$ as follows:

$G_0 = G$;

$G_{\alpha+1} = (X_{\alpha+1}, E_{\alpha+1})$, where $X_{\alpha+1} = X_{\alpha} \cup D(G_{\alpha})$, and $E_{\alpha+1} = E_{\alpha}$ together with edges of the form $(x,X)$, where $x\in X_{\alpha}$, $X \in D(G_{\alpha})$, and $x\in X$.

For limit $\alpha$, $G_\alpha$ is the union of $G_\beta$ for $\beta<\alpha$.

The model/digraph we are interested in is the union of all the $G_\alpha$, as $\alpha$ ranges over the ordinals, and $G$ is the 3-element digraph on {$a,b,c$} mentioned earlier. Let's call this model $V(a,b,c)$. It satisfies all the axioms of $ZF$ with the exception of Foundation.

$V(a,b,c)$ also satisfies $S$ since any infinite descending epsilon chain must eventually hit $a$, $b$, or $c$.

So this shows that Question 3 has a negative answer.

Since {$a,b,c$} are indiscernibles in $V(a,b,c)$, I suspect that $DC$ fails in $V(a,b,c)$, but a variation on this theme might produce a model with enough asymmetry for $DC$ to hold as well.

PS. Models of $ZF$ in which the Foundation fails are often constructed using the so-called Bernays-Rieger permutation method (not to be confused with the Fraenkel-Mostowski permutation method of constructing models of $ZF$ in which the axiom of choice fails). The model constructed above is based on a different idea, explored in detail for models of finite set theory in the following paper:

A. Enayat, J. Schmerl, and A. Visser, Omega Models of Finite Set Theory , to appear in Set theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies (edited by J. Kennedy and R. Kossak), Cambridge University Press, to appear October 2011.

A preprint can be found here.

Since Joel has already answered Questions (1), and (2), I will only offer an answer for Question 3.

Start with 3 "urelements" {$a,b,c$} and define the von Neumann universe on top of them [keep taking power sets, and take unions at limits, so the construction has length Ord]. Call this model $V(a,b,c)$, it satisfies all the axioms of $ZF$ with the exception of Extensionality and Foundation.

$V(a,b,c)$ also satisfies $S$ since any infinite descending epsilon chain must eventually hit $a$, $b$, or $c$.

To gain Extensionality, we extend the epsilon relation for $V(a,b,c)$ [thinking graph-theoretically] by adding the epsilon "edges" from each of {$a,b,c$} to the other two, so that the epsilon relation on {$a,b,c$} becomes a simple graph on 3 vertices. This addition of edges to $V(a,b,c)$ gives rise to a new model $V^*(a,b,c)$, which continues to satisfy $S$ and all of $ZF$ with the exception of Foundation.

So this shows that Question 3 has a negative answer.

Since {$a,b,c$} are indiscernibles in $V(a,b,c)$, $DC$ fails in $V(a,b,c)$, but a variation on this theme might produce a model with enough asymmetry for $DC$ to hold as well.

Since Joel has already answered Questions (1), and (2), I will only offer an answer for Question 3.

This is a revised version of my answer; thanks to Joel Hamkins for pointing out that my previous construction was not quite right.

Start with a with a simple graph with 3 elements {$a,b,c$}, where each of the three nodes has an edge to the other two. So in this 3-element model of "set theory", $a$ = {$b$, $c$}, $b$ = {$c$, $a$}, and $c$ = {$a$, $b$}.

Given a digraph $G=(X,E)$, with $X$ as the vertex set and $E$ as the edge set, define the deficiency set $D(G)$ of $G$ to be the collection of subsets $S$ of $X$ that are not "coded" in $G$, i.e., there is no element $a$ in $X$ such that $S$ = {$x \in X : xEa$}.

We now can define by recursion a digraph $G_\alpha = (X_\alpha, E_\alpha)$ for each ordinal $\alpha$ as follows:

$G_0 = G$;

$G_{\alpha+1} = (X_{\alpha+1}, E_{\alpha+1})$, where $X_{\alpha+1} = X_{\alpha} \cup D(G_{\alpha})$, and $E_{\alpha+1} = E_{\alpha}$ together with edges of the form $(x,X)$, where $x\in X_{\alpha}$, $X \in D(G_{\alpha})$, and $x\in X$.

For limit $\alpha$, $G_\alpha$ is the union of $G_\beta$ for $\beta<\alpha$.

The model/digraph we are interested in is the union of all the $G_\alpha$, as $\alpha$ ranges over the ordinals.

This model, let's call it $V(a,b,c)$, satisfies all the axioms of $ZF$ with the exception of Foundation.

$V(a,b,c)$ also satisfies $S$ since any infinite descending epsilon chain must eventually hit $a$, $b$, or $c$.

So this shows that Question 3 has a negative answer.

Since {$a,b,c$} are indiscernibles in $V(a,b,c)$, $DC$ fails in $V(a,b,c)$, but a variation on this theme might produce a model with enough asymmetry for $DC$ to hold as well.

PS. Models of $ZF$ in which the Foundation fails are often constructed using the so-called Bernays-Rieger permutation method (not to be confused with the Fraenkel-Mostowski permutation method of constructing models of $ZF$ in which the axiom of choice fails). The model constructed above is based on a different idea, explored in detail for models of finite set theory in the following paper:

A. Enayat, J. Schmerl, and A. Visser, Omega Models of Finite Set Theory , to appear in Set theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies (edited by J. Kennedy and R. Kossak), Cambridge University Press, to appear October 2011.

A preprint can be found here.