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.