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.