Nicolas Destainville [arXiv:cond-mat/0101413] suggests that the "coupling from the past" algorithm can efficiently produce an exactly uniformly random rhombus tiling of any zonogon with integer side lengths.
There is no general formula because it is understood that the number isn't "round" (i.e., a product of small factors, especially one that potentially yields a $q$-analogue). However, it was discovered experimentally that the number of tilings of an $(a,b,1,1)$ octagon is round, and this was proven by Elnitsky in the reference that you cited.
An entropy model of the tilings, including the question of just approximating the number of tilings, is a really good question that I suspect is open. My evidence for this is that the general entropy study for hexagons is non-trivial even though it is solved. The full answer is that a random tiling has an inscribed Arctic ellipse with saturated entropy only in the very the center. I guess the first important paper on this ellipse is the one by Cohn, Larsen, and Propp [arXiv:math/9801059].
There is another interesting interpretation of these tilings (which you might already know). They are discrete minimal surfaces in the cubical lattice in $\mathbb{R}^g$. You can use this interpretation to prove that they are all connected by hexagon moves.
Finally, a pet peeve: The Kasteleyn method is not fundamentally different from the Gessel-Viennot method [arXiv:math/9810091]. There will never be a counting problem in which one method applies and some version of the other method does not. The matrices of the two methods are always equivalent in the sense of $K$-theory, i.e., up to integer changes of basis and stabilization.