Since B-fields are being discussed in the other question, i'll try to answer your question on the complexified Kahler moduli. The set of Kahler classes of a Kahler manifold is an open cone in $H^{1,1}(X, C) \cap H^2(X, R)$. There are results on the structure of this cone which roughly say that it is rational polyhedral away from some set (I think!). In the construction of the complexified Kahler cone you have to take a "framing" of this cone, in the end you get something isomorphic to $(\Delta^{\ast})^{h^{1,1}}$, where $\Delta^*$ is the punctured disc in $C$. This matches the moduli of complex structures of the mirror, near a large complex structure limit point. These things are discussed in the book "Calabi-Yau manifolds and related geometries", by Gross, Huybrechts and Joyce.
Maybe I should comment. The short answer is "Hopefully all the way", but there are some caveats. Our program indeed started out by the observation that from a physical reasoning mirror symmetry for Calabi-Yau varieties only works near degeneration limits. The reason is that while the topological B-model (the complex side) works for any Calabi-Yau variety, the topological A-model (the symplectic side) becomes unreliable for small Kähler classes. As one example of a mathematical manifestation of this there are Calabi-Yau manifolds with two different maximal degenerations leading to non-deformation equivalent mirrors, such as the Pfaffian Calabi-Yau (http://arxiv.org/abs/math/9801092). Another mathematical manifestation already mentioned by Scott Carnahan is the fact that the Fukaya category is only defined over the Novikov ring and at best converges for large symplectic forms. One might phantasize about a strict analogue of the complex moduli space on the mirror side, a stringy Kähler moduli space, but at present it is not clear what this should be. Going over to the homological point of view does not appear to help directly either, as the example of the Fukaya category shows.
The point I want to make is that if you want a statement staying within the realm of Calabi-Yau varieties and Fukaya categories I don't see a way around a perturbative formulation. There are nevertheless non-perturbative manifestations of mirror symmetry and powerful non-perturbative computational techniques. For example, the global structure of the complex moduli space has been repeatedly put to use with great effect, e.g. to solve the holomorphic anomaly equation. We have no means to see this input from the global geometry of the complex moduli space perturbatively, and this is probably what Mark meant in his answer in Michigan in 2008 (but see below for recent progress on this via theta functions). The one exception I am aware of is the recent Chiodo-Ruan-fantasy about "global mirror symmetry". While I haven't thought deeply about this, the examples are about hypersurfaces, and these can be studied as Landau-Ginzburg models by working with the homogeneous equations on affine space (LG/CY-correspondence). Landau-Ginzburg models fit into our program as well, see my paper with Michael Carl and Max Pumperla (on my webpage, yet to be polished), so hopefully this has an interpretation from our point of view as well.
As for the question on a relation to homological mirror symmetry, we are just about to finish a survey "Theta functions and mirror symmetry" that sketches how we imagine to prove homological mirror symmetry via tropical Morse trees. The point is that our construction comes with a canonical basis of sections ("theta functions") of powers of the ample line bundle. These are mirror dual to intersection points of a class of Lagrangian sections of an SYZ fibration, viewed in the degeneration limit. For the elliptic curve Mark has worked out the correspondence in great detail in Chapter 8 of the book "Dirichlet branes and mirror symmetry". The theta functions, by the way, often turn out to be algebraic and may help to probe deeper into the moduli space, but this is still to be understood.
Tropical curves should also be at the heart of the correspondence between Gromov-Witten invariants and periods that you were also wondering about. There are some loose ends here, but Mark's paper "Mirror symmetry for P^2 and tropical geometry" and our joint paper with Rahul Pandharipande should give an impression of the picture.
Best Answer
Since no one else has tried to answer, I'll take a shot. It seems to me that there are threads of ideas in this story that in the very distant future might be woven together to give a possible answer.
To begin, we should note that there seems to be a general idea, discussed in this mathoverflow question, Deformation quantization and quantum cohomology (or Fukaya category) -- are they related?. That one could define the Fukaya category as modules over a deformation quantization of $C^{\infty}(X)$ corresponding to the symplectic form $\omega$.
The basic idea is that in two naive respects this category of modules behaves a lot like the Fukaya category. Firstly, the Hochschild cohomology of the deformation quantization is almost by definition the Poisson cohomology of the symplectic form $\omega$, which in turn is known to be isomorphic to $H^*(X)((t))$. As an equation:
$$HH^*(A_\omega,A_\omega) \cong H^*(X)((t)) $$
Second, one can define a reasonable notion of modules with support on a Lagrangian submanifold and for any Lagrangian L, produce canonical holonomic modules supported there. One can compute that $$Ext(M_L,M_L) \cong H^*(L)((t))$$ There is some hope that one can put in the instanton corrections in a formal algebraic way and a fair amount of work has been done in this direction.
This story works best so far for the Fukaya category of $T^*X$ where the deformation quantization is roughly the algebra of differential operators. This is related to more work than I could competently summarize. I'll just mention, work of Nadler and Zaslow, Tsygan and Tamarkin. This approach is used by Kapustin and Witten to incorporate co-isotropic branes into the Fukaya category in their famous study of the Geometric Langlands. There, they are after some enlargement of Nadler's infinitesimal Fukaya category of $T^*(X)$. Note however that this not the same Fukaya category(the wrapped Fukaya category) that one studies in the context of mirror symmetry, but perhaps things will work better in the compact case if that is ever put on firm ground.
This was all a prelude to say that deformation quantization places you firmly in the land of non-commutative geometry anyways. Things like differential operators for non-commutative rings can make sense http://www.springerlink.com/content/r0rqguawu1960qxy/. I've never really looked at Van Den Bergh's work, but perhaps the passage from the sheaf of algebraic functions to the sheaf $C^\infty(X)$ is another stumbling point. One of Maxim's Kontsevich's ideas (see his Lefschetz lecture notes http://www.ihes.fr/~maxim/TEXTS/Kontsevich-Lefschetz-Notes.pdf) is that for any saturated dg-algebra there should maybe exist some nuclear algebra which bears the same formal relationship as the algebra of algebraic functions and smooth functions.