I believe that an answer to your question [1] is the system that Dana Scott developed in his paper, "Axiomatizing Set Theory" found in Proceedings of Symposia in Pure Mathematics, Volume 13, Part II, 1974, pp. 207-14. This system is, in Scott's words, the formalization of the following intuition:
But note that our original intuition of set is based on the idea of having collections of already fixed objects. The suggestion of considering all-inclusive collections [such as the set of all sets not containing itself, the set of all ordinals, etc.--my comment] only came in later by way of formal simplification of language. The suggestion proved unfortunate, and so we must return to the primary intuitions. These intuitions can gain an initial precision through formulating the two basic axioms of extensionality and comprehension, which we now discuss in detail.
Scott first, however, develops the following, "nameless", axiom:
Let the variables $a$, $b$, $c$, $a^{'}$, $b^{'}$, $c^{'}$, $a^{''}$,... range over sets and the variables $x$, $y$, $z$, $x^{'}$, $y^{'}$, $z^{'}$, $x^{''}$... range over arbitrary objects. The symbol $=$ is used for identity and $\in$ for membership. Whether it is really interesting or profitable to allow for non-sets in the theory is debatable; but let us not exclude them yet. We agree that the condition $x$ $\in$ $y$ should imply that $y$ is a set, a principle that we can formulate in logical symbols thus:
$\forall$$x$,$y$[$x$ $\in$ $y$ $\rightarrow$ $\exists$$a$[$y$ = $a$]]
Of course this must be taken as a axiom; but it is so primitive, so much just a convention of grammar, that we will not even give it a name.
Scott then gives the axioms of extensionality and comprehension:
Extensionality. $\forall$$a$,$b$[$\forall$$x$[$x$ $\in$ $a$ $\leftrightarrow$ $x$ $\in$ $b$] $\rightarrow$ $a$ = $b$].
Comprehension. $\forall$$a$ $\exists$$b$$\forall$$x$[$x$ $\in$ $b$ $\leftrightarrow$ $x$ $\in$ $a$ $\land$ $\Phi$($x$)].
Regarding Extensionality, Scott writes:
The extensionality axiom formalizes our idea that a set is nothing more than a collection of objects: It is uniquely determined by its elements.
Regarding the Comprehension axiom, Scott has a bit more to say:
The comprehension axiom formalizes the idea that one a collection $a$ is fixed, we can then extract from $a$ any arbitrary subcollection $b$. The extraction process is effected by finding a property $\Phi$($x$) which distinguishes $b$ as the subset of $a$ comprehending all those elements having the property. There is no reason to place any restriction on how $\Phi$($x$) is formulated: We believe in the existence of arbitrary subsets. It is a great temptation to erase the condition $x$ $\in$ $a$, thus simplifying the axiom schema; but we all know what happens. It is much more profitable to ask: Where does the $a$ come from?
He then proceeds to discuss (from a historical perspective) the answer to the above question:
Zermelo answered the question by giving several construction principles for obtaining new $a$'s from old. Fraenkel and Skolem extended the method, and von Neumann, Bernays and Godel modified it somewhat. Actually this is a rather sad history--because set theory is made to seem so artificial and formalistic. The naive axioms are contradictory. We block the contradiction and thereby emasculate the theory. Therefore to get anywhere we reinstate a few of the principles we eliminated and hope for the best. Now it would be wrong to accuse any of the above men of holding such a simplistic view of the axiomatic process. Nevertheless it is a widely held view and one that is easy to fall into when considering the formal axioms. Let us try to see whether there is another path to the same theory more obviously based on the underlying intuition.
Scott the goes on to make the following claim:
The truth is that there is only one satisfactory way of avoiding the paradoxes: namely, the use of some form of the theory of types . That was the basis of of both Russell's and Zermelo's intuitions. Indeed the best way to regard Zermelo's theory is a simplification and extension of Russell's. (We mean Russell's simple theory of types, of course.) Thus mixing of types is easier and annoying repititions are avoided. Once the later types are allowed to accumulate the earlier ones, we can then easily imagine extending the types into the transfinite--just how far we want to go must necessarily be left open. Now Russell made his types explicit in his notation and Zermelo left them implicit. It is a mistake to leave something so important invisible, because so many people will misunderstand you. What we shall try to do here is to axiomatize the types in as simple a way as possible so that everyone can agree that the idea is natural.
He now provides the basic intuitions for his "cumulative types":
Let us proceed on a very primitive basis. In order to obtain the sets to which the comprehension axioms can be applied, we imaginesome way of dividing the sets into levels. There will be earlier levels and later levels. Let the sets up to a certain level be thought of as forming a partial universe $V$ which is regarded as a legitimate set [in Zuhair's terminology, a small set--my comment]. We can be generous and assume that all the nonsets, the set-theoretic atoms, belong to all the levels. In a later universe $V^{'}$ we have not only the elements of $V$, but also $V$ itself to be used to form subcollections of $V^{'}$; that is, $V$ $\in$ $V^{'}$ as well as $V$ $\subseteq$ $V^{'}$, where we define
$\forall$$x$,$y$[$x$ $\subseteq$ $y$ $\leftrightarrow$ $\forall$$z$[$z$ $\in$ $x$ $\rightarrow$ $z$ $\in$ $y$]].
Furthermore, not only $V$, but by the same token all the subcollections of $V$, should also be elements of $V^{'}$.
Scott continues defining the notion of type in his system:
Once a set is fixed at one level, all its subsets are fixed at a later level--that is certainly the basic idea of the theory of types. We formalize this idea not by introducing type indices, but more simply by identifying a level with the collection of all sets (and nonsets) up to that level. We let variables $V$, $V^{'}$, $V^{''}$,... range over these levels--that is, we take the idea of a type level (as identified with certain sets) as a primitive notion. The "later than" relation is transcribed simply as $V$ $\in$ $V^{'}$. It need hardly be mentioned that we assume that there is at least one level and that each level is a set--axioms that we do not stop to name. What is important is the idea that a given level is nothing more than the accumulation of all the members and subsets of all the earlier levels (and all the nonsets, if any there be). In formal terms we have this axiom:
Accumulation. $\forall$$V^{'}$$\forall$$x$[$x$ $\in$ $V^{'}$ $\leftrightarrow$ $\lnot$$\exists$$a$[$x$=$a$] $\lor$ $\exists$$V$ $\in$ $V^{'}$[$x$ $\in$ $V$ $\lor$ $x$ $\subseteq$ $V$]].
(By the way, just because we use the variables $V$, $V^{'}$, $V^{''}$,... we should not think of the levels as being arranged in a $\omega$-type sequence. In general we will want a transfinite sequence. Also note that $V$ $\in$ $V^{'}$ does not imply that $V^{'}$ is the next level; it may be a much later level.) The purpose of this axiom is to show how the levels fit together.
The next axiom captures the intuition that however far the levels go out, they eventually capture everything:
Restriction. $\forall$$x$$\exists$$V$[$x$ $\subseteq$ $V$]
In other words, the whole universe, if only it were a set, would behave as the ultimate level in the sense of the previous axiom. (Note that this axiom gives the existence of at east one level. It really should have been formulated with the clause [$x$ $\in$ $V$ $\lor$ $x$ $\subseteq$ $V$], but we shall show below that $x$ $\in$ $V$ $\rightarrow$ $x$ $\subseteq$ $V$].) This will turn out to be nothing more or less than the well-known axiom of foundation, which is generally poorly understood. We feel that in the present context, it appears as a quite natural expression of the fact that the sets are restricted to levels.
Finally, Scott includes the following reflection principle as the last axiom:
Reflection. $\exists$$V$$\forall$$x$$\in$$V$[$\Phi$($x$) $\rightarrow$ $\Phi^{(V)}$($x$)] [from which one can derive infinity and replacement--my comment. See pp.213-214].
It should be noted that Scott, from Extensionality, Comprehension, Accumulation, and Restriction, shows that
(union and power set) also drop out of these axioms [pp. 210-212--my comment]
and that by Restriction, one can add complements over the whole domain of discourse so the existence of Scott's system is a correct answer to question [1].
