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.