Moduli spaces of pseudoholomorphic curves do not carry the structure of a (compact) differentiable manifold in general (due to transversality issues). Nevertheless one would like to at least associate a fundamental class to the moduli space in question.
It looks like two approaches dominate: Kuranishi structures and polyfolds.
Both seem to be mammoth projects. Before diving seriously into one of them I may ask:
What are the advantages/drawbacks of these approaches? What is their motivation? Are they rigourosly proved? Are there reasonable alternatives?
[Math] Kuranishi structures vs polyfolds
sg.symplectic-geometry
Related Solutions
Here's a view of the symplectic side of the bridge.
The Kuranishi model (see Donaldson-Kronheimer, The geometry of four-manifolds, ch. 4) goes like this. You're interested in a (moduli) space $M$ cut out as $\psi^{-1}(0)$, for some smooth but nonlinear map of Banach spaces, $\psi \colon (E,0) \to (F,0)$ such that $\delta:=D_0\psi$ is a Fredholm operator (finite dim kernel and cokernel). That means that $\delta$ is "almost" an isomorphism, and Kuranishi's principle is that one can construct a non-linear map $\kappa \colon \ker(\delta) \to \mathrm{coker}(\delta)$, such that $\kappa(0)=0$ and $D_0 \kappa =0$, and (locally near $0$) a homeomorphism $M \to \kappa^{-1}(0)$. This gives a finite-dimensional model of $M$. "Kuranishi structures" are a formalism in which one can say that $M$ is everywhere-locally given as the zeros of maps like $\kappa$.
In the case of genus 0 GW theory, $M$ is the moduli space of (say) parametrized pseudo-holomorphic maps from $S^2$ to an almost complex manifold $X$; $\psi$ is a non-linear Cauchy-Riemann operator, and, for a pseudo-holomorphic map $u\in M$, $D_u \psi$ is a linearized C-R operator - the $(0,1)$-part of a covariant derivative acting on sections of $u^\ast TX$. Its kernel can be identified with the holomorphic sections $H^0(S^2,u^\ast TX)$ of the holomorphic structure on the vector bundle $u^\ast TX$ defined by the C-R operator. Its cokernel is isomorphic to $H^1(S^2,u^\ast TX)$. If you're lucky, you have a Zariski-smooth moduli space $M$ whose Zariski tangent space at $u$ is $\ker D_u\psi$. In this case, one could take $\kappa=0$, and then the spaces $\mathrm{coker} (D_u\psi)$ form a vector bundle $Obs \to M$, which is what symplectic geometers usually call the obstruction bundle. One can now try to divide everything by $Aut(S^2)$, and quite possibly get into orbi-mathematics.
In the integrable case, still with non-singular but excess-dimensional $M$, I could write $Obs$ as $R^1\pi_* \Phi^*T_X$, where $T_X$ is the holomorphic tangent sheaf, $\Phi$ the evaluation map $\mathbb{P}^1\times M \to X$, and $\pi$ the projection to $M$. At this point, if what I've said is accurate, you and other algebraic geometers out there are better placed than me to answer your question. Is it your $(E_{-1})^\vee$?
Of course, you didn't really want to assume $M$ smooth. A place where these deformation theories are compared in generality is Siebert's 1998 paper Algebraic and symplectic Gromov-Witten invariants coincide.
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
Kuranishi models are a traditional - and beautiful - technique for describing the local structure of moduli spaces cut out by non-linear equations whose linearization is Fredholm. A more elaborate version, "Kuranishi structures", are used by Fukaya-Oh-Ohta-Ono (FOOO) and Akaho-Joyce to handle transversality for moduli spaces of pseudo-holomorphic polygons with Lagrangian boundary conditions, and the compactifications of these spaces. FOOO's book is the result of a decade of dedicated thought by a superb team, but few have assimilated it (I certainly haven't).
Polyfolds, Hofer's "infrastructure project", are designed with the severe demands of symplectic field theory (SFT) foremost in mind. This is a more radical rethink of how to handle transversality, whose aims include absorbing the difficulties of the lack of canonical coordinates when gluing Morse-Floer trajectories. (What difficulties? Try to prove that the moduli space of unparametrized broken gradient flow-lines of an ordinary Morse function is a smooth manifold with corners, and you'll find out.) Several papers into the project, it's still not completely clear how efficiently it will work in applications, especially those outside SFT. I'd hope that polyfolds will help us set up Cohen-Jones-Segal Floer homotopy-types, for instance - but there may still be severe difficulties. I've also never heard a compelling argument that Kuranishi structures are insufficient for SFT.
This paper of Cieliebak-Mohnke suggests an intriguing alternative approach.
My view would be that these mammoths are worth chasing only if you have very clear aims in mind. There are many excellent problems in symplectic topology that don't need such gigantic foundations. If you're interested in Fukaya categories, there's lots to be proved using the definition from Seidel's book, which deals with exact symplectic manifolds. If you want to prove things about contact manifolds, try using symplectic cohomology, a close cousin of SFT requiring less formidable analysis.