Ian Morrison wrote up some nice lectures in the book Lectures on Riemann surfaces,World Scientific publishers, Proceedings of the college of Riemann surfaces in 1987, at the ICTM in Trieste. They were intended as an informal introduction to the two detailed treatments mentioned below by Mumford (l'Enseignement) and Gieseker (Tata).
There are notes on Ravi Vakil's web page for his course on deformations and moduli:
http://math.stanford.edu/~vakil/727/index.html
There is a nice treatment of the chow coordinates of a projective variety in chapter 1 of the book Basic algebraic geometry by Shafarevich. This is very elementary and readable.
There is a good discussion of the existence of the Hilbert scheme in Mumford's book Lectures on curves on an algebraic surface, Annals of math studies #59. Sophisticated, but we were able to use it in a seminar long ago, and got some good insight from it.
Mumford (notes by Morrison) first wrote up the case of stable curves in Stability of projective Varieties, in l'Enseignement mathematique, 1977, based on an idea of Gieseker.
Then Gieseker himself presented his version at the Tata Institute in Bombay (TIFR), and wrote it up in their series of lectures on mathematics and physics, #69, 1982.
The original presentation of the concept of stable curves, due to Alan Mayer and David Mumford, is in talks by Mayer and Mumford at the Woods Hole conference 1964, available on James Milne's web site at Michigan, or that of roy smith (mathwonk) at University of Georgia.
As I recall, even the detailed works by Mumford, (GIT, Enseignement), always include some introductory examples and motivation that anyone can read, so one should not shy away from the actual definitive works completely. In regard to the fine recommendations above, Mukai's is actually a textbook as requested, and not a monograph like most of my recommendations here, but that of course makes it longer.
For beginners, I would observe that the Chow approach is to characterize a projective variety by all lines meeting it, thus getting a subset of the Grassmannian of lines, while the Hilbert approach is to describe a variety by the set of all hypersurfaces of fixed large degree containing it, thus getting a subspace of the vector space of those polynomials, another Grassmannaian. Then to characterize abstract varieties, one first chooses some natural projective embedding, say by a multiple of the canonical class, then considers the corresponding Hilbert or Chow scheme, and tries to collapse together all different embeddings of the same variety, in GIT by taking a quotient by a group action. This then leads to singularities at orbits which are smaller than usual, i.e. at points with non trivial isotropy coming from automorphisms of the variety. These isotropy groups are included in the data of a moduli "stack", but were always considered informative even earlier.
Since the subject is huge, it helps also to know which aspect is of interest. A moduli space is usually the set of isomorphism classes of objects of a given type. Hence, aside from foundational subtleties, it “exists” as a set. Then the problem is to give it more structure and to prove it has some nice properties. In algebraic geometry one often tries to give it structure as an algebraic space, scheme, or quasi projective variety, perhaps progressively in that order. So the first job would be to define a natural structure as abstract topological space or even abstract scheme. Next one wants to capture this structure by some “moduli”.
Classically, “moduli” are numbers that distinguish non isomorphic objects, i.e. numerical invariants such as projective coordinates in a field, so this translates into the stage of giving a structure of quasiprojective variety. This requires finding embedding functions, or sections of line bundles which are constant on equivalence classes. If the equivalence classes are orbits of a group action, one seeks functions constant on orbits, i.e. “invariants”, and this is the subject of “invariant theory".
Since algebraic projective mappings are continuous, their level sets are closed, so the geometric invariant theory problem arises of which orbits are closed. This leads into various concepts of “stability” of objects under a given action, and also, since closure is a relative notion, of determining certain unstable subsets to exclude so that the remaining orbits become closed. This is the subject studied by Mumford in which he adapted ideas of Hilbert.
Finally, one wants to find a good geometric compactification of the given moduli space, since the set of isomorphism classes of a given type is seldom compact. The method of parametrizing moduli spaces by subsets of Hilbert schemes, yields a natural compactification, since Hilbert schemes are projective, but since all isomorphism classes of the original type were already present before compactifying, it is unclear what geometric objects the new points added correspond to. This leads to the challenge of identifying the Hilbert scheme compactification with a more abstract compactification which adds in degenerate versions of the original geometric objects. These abstract objects are called perhaps “moduli stable” objects of the orginal kind, and one must show this abstract compact space can be identified with some version of the Hilbert scheme projective compactified one.
The concept of (moduli) stable curves was introduced by Mayer and Mumford, and the next job was to show they give a good abstract separated compactifiction of M(g). This is presumably the content of the paper of Deligne and Mumford. Then the proof they in fact give a natural projective compactification in the Hilbert scheme GIT sense is apparently accomplished in the references of Mumford and Gieseker.
Aside from these global aspects of moduli there are local questions, such as what is the dimension of a (component of a) moduli space, or what is its tangent space? These are the concern of “deformation theory”, or the local variations of structure of a given object. Here also one distinguishes deformations of the original objects, usually non singular varieties or manifolds, as in the works of Kodaira, from deformations of the degenerate objects included at the boundary of the compactification, i.e. deformations of singularities. For the latter there is a nice Tata lecture note by M. Artin, and a recent book by Greuel, Lossen, and Shustin. All sources rely fundamentally on the unpublished 1964 PhD thesis of M. Schlessinger at Harvard.
After all these foundations are settled, it remains to compute invariant properties of the resulting moduli spaces, their singularities, canonical class, Kodaira dimension, Picard group, cohomology, chow ring, rational curves in them,….. For M(g)bar this is still going in progress.
However it seems to me most answers, especially mine, are oriented to geometric questions as opposed to the requested arithmetic ones. Should one suggest some works say by Faltings and Chai?
Question 1 (compare virtual fundamental cycles of different perfect obstruction theories on space underlying space): There is essentially no relation between $[X]_\varphi$ and $[X]_{\varphi'}$ for different perfect obstruction theories $\varphi:E^\bullet\to\mathbb L_X$ and $\varphi':E^{\prime\bullet}\to\mathbb L_X$. The "derived" structure on $X$ encoded in $\varphi$ is essential for defining the vfc; knowing $X$ as a topological space (or variety, stack, etc.) determines essentially nothing about vfc (except trivial things like the fact that the vfc vanishes if the virtual dimension is larger than the classical dimension of $X$).
A very special case of your question is "Does the euler class $e(E)\in H^\bullet(M)$ of a vector bundle $E$ over a manifold $M$ depend only on $M$?" whose answer should be clear.
Question 2 (axiomatic characterization of virtual fundamental classes): I've thought extensively about this problem, and as far as I know, no axiomatic characterization of virtual fundamental classes/cycles/chains has been formulated and proved in the literature. There is definitely no satisfying general result which allows one to compare all reasonable approaches to defining virtual fundamental cycles in symplectic geometry. The philosophical reason why this seems like a difficult problem is that it's much easier to work "infinitesimally" in algebraic geometry than in differential geometry (or, at least, the sort of differential geometry relevant to moduli spaces of pseudo-holomorphic curves). Thus, all existing methods for defining the VFC in symplectic geometry "remember" much more of the ambient geometry of the entire space of smooth (as opposed to pseudo-holomorphic) maps than should be necessary for defining the VFC. The comparison between them is very technical because, although morally all approaches give rise to exactly the same VFC, we don't currently have a good language for recording the minimal amount of "derived" information that the moduli spaces carry (and which should be sufficient for defining the VFC).
Ideally, one would like to define some (derived?) moduli problem in the smooth or topological category for pseudo-holomorphic curves. Then one would like to show that this moduli problem is representable by a reasonable "derived topological manifold" (or orbifold) whose underlying topological space is the usual moduli space and whose derived structure is the analogue of a perfect obstruction theory. The last step (and probably the easiest, actually) is defining the VFC from this derived structure.
In my view, an axiomatic characterization of virtual fundamental classes is unlikely to be helpful with the question of comparing different constructions in symplectic geometry, unfortunately. This is simply because the "problem" is more than just having various ways of extracting the VFC, rather it's that we don't even know what the right canonical extra "derived" structure on the moduli space is from which we should extract the VFC. I'd be thrilled if I'm wrong, though!
I'll stop here, although it's possible to write endlessly on this topic. If you have other questions, I'm happy to expand this answer or answer a subsequent question you ask.
Best Answer
Dick Hain recently released a very nice set of lecture notes which are available here :
Lectures on moduli spaces of elliptic curves
They don't cover the precise list of topics you list, but they are quite readable.