Well the first thing to say is to look at the very enthusiastic and world-encompassing papers of Cherednik himself on DAHA as the center of the mathematical world (say his 1998 ICM).
I'll mention a couple of more geometric aspects, but this is a huuuge area..
There are at least three distinct geometric appearances of DAHA, which you could classify by the number of loops (as in loop groups) that appear - two, one or zero.
(BTW for those in the know I will mostly intentionally ignore the difference between DAHA and its spherical subalgebra.)
Double loop picture: See e.g. Kapranov's paper arXiv:math/9812021 (notes for lectures of his on it available on my webpage) and the related arXiv:math/0012155. The intuitive idea, very hard to make precise, is that DAHA is the double loop (or 2d local field, such as F_q((s,t)) ) analog of the finite (F_q) and affine (F_q((s)) ) Hecke algebras. In other words it appears as functions on double cosets for the double loop group and its "Borel" subalgebra. (Of course you need to decide what "functions" or rather "measures" means and what "Borel" means..) This means in particular it controls principal series type reps of double loop groups, or the geometry of moduli of G-bundles on a surface, looked at near a "flag" (meaning a point inside a curve inside the surface). The rep theory over 2d local fields that you would need to have for this to make sense is studied in a series of papers of Kazhdan with Gaitsgory (arXiv:math/0302174, 0406282, 0409543), with Braverman (0510538) and most recently with Hrushovski (0510133 and 0609115). The latter is totally awesome IMHO, using ideas from logic to define definitively what measure theory on such local fields means.
Single loop picture: Affine Hecke algebras have two presentations, the "standard" one (having to do with abstract Kac-Moody groups) and the Bernstein one (having to do specifically with loop groups). These two appear on the two sides of Langlands duality (cf eg the intro to the book of Chriss and Ginzburg). Likewise there's a picture of DAHA that's dual to the above "standard" one. This is developed first in Garland-Grojnowski (arXiv:q-alg/9508019) and more thoroughly by Vasserot arXiv:math/0207127 and several papers of Varagnolo-Vasserot. The idea here is that DAHA appears as the K-group of coherent sheaves on G(O)\G(K)/G(O) - the loop group version of the Bruhat cells in the finite flag manifold (again ignoring Borels vs parabolics). Again this is hard to make very precise. This gives in particular a geometric picture for the reps of DAHA, analogous to that for AHA due to Kazhdan-Lusztig (see again Chriss-Ginzburg).
[EDIT: A
new survey on this topic by Varagnolo-Vasserot has just appeared.]
Here is where geometric Langlands comes in: the above interp means that DAHA is the Hecke algebra that acts on (K-groups of) coherent sheaves on T^* Bun_G X for any Riemann surface X -- it's the coherent analog of the usual Hecke operators in geometric Langlands.
Thus if you categorify DAHA (look at CATEGORIES of coherent sheaves) you get the Hecke functors for the so-called "classical limit of Langlands" (cotangent to Bun_G is the classical limit of diffops on Bun_G).
The Cherednik Fourier transform gives an identification between DAHA for G and the dual group G'. In this picture it is an isom between K-groups of coherent sheaves on Grassmannians for Langlands dual groups (the categorified version of this is conjectured in Bezrukavnikov-Finkelberg-Mirkovic arXiv:math/0306413). This is a natural part of the classical limit of Langlands: you're supposed to have an equivalence between coherent sheaves on cotangents of Langlands dual Bun_G's, and this is its local form, identifying the Hecke operators on the two sides!
In this picture DAHA appears recently in physics (since geometric Langlands in all its variants does), in the work of Kapustin (arXiv:hep-th/0612119 and with Saulina 0710.2097) as "Wilson-'t Hooft operators" --- the idea is that in SUSY gauge theory there's a full DAHA of operators (with the above names). Passing to the TFT which gives Langlands kills half of them - a different half on the two sides of Langlands duality, hence the asymmetry.. but in the classical version all the operators survive, and the SL2Z of electric-magnetic/Montonen-Olive S-duality is exactly the Cherednik SL2Z you mention..
Finally (since this is getting awfully long), the no-loop picture: this is the one you referred to in 2. via Dunkl type operators. Namely DAHA appears as difference operators on H/W (and its various degenerations, the Cherednik algebras, appear by replacing H by h and difference by differential). In this guise (and I'm not giving a million refs to papers of Etingof and many others since you know them better) DAHA is the symmetries of quantum many-body systems (Calogero-Moser and Ruijsenaars-Schneiders systems to be exact), and this is where Macdonald polynomials naturally appear as the quantum integrals of motion.
The only thing I'll say here is point to some awesome recent work of Schiffmann and Vasserot arXiv:0905.2555, where this picture too is tied to geometric Langlands..
very very roughly the idea is that H/W is itself (a degenerate version of an open piece of) a moduli of G-bundles, in the case of an elliptic curve. Thus studying DAHA is essentially studying D-modules or difference modules on Bun_G in genus one (see Nevins' paper arXiv:0804.4170 where such ideas are developed further). Schiffman-Vasserot show how to interpret Macdonald polynomials in terms of geometric Eisenstein series in genus one..
enough for now.
Perhaps this, for now, is more an issue of perspective. Yes, for matroids, spheres and Coxeter groups the realizable cases were known before using results in algebraic geometry, but this is natural as our understanding of the cohomology of algebraic varietes was much better, historically. And so we think of this as strange because we are used to think of this in terms of algebraic varieties.
However, matroids, for instance, are perhaps more naturally thought of in the context of valuations, and there, it suddenly becomes more natural to consider McMullen's argument for the Lefschetz theorem and the Hodge-Riemann relations (and this is ultimately what is used).
Similarly, spheres are rarely ever polytopal, and even for those that are, the realization as a polytope is an unnatural straightjacket. We do, however, understand them well in terms of cobordisms, and we do know general position tricks from when we define intersection products in cohomology. And this ultimately leads to the Lefschetz theorem there.
Best Answer
The phenomenon seems the same as for affine Kac-Moody algebras, where rational level is where everything special happens, or quantum groups, where roots of unity are the exceptional locus (and of course these examples are related). The parameter for Cherednik algebras is an additive/Lie algebra type parameter, like the KM level, as opposed to the exponentiated version as in the quantum groups case. If you mod out by the action of translation functors, you're asking why finite order points of the parameter are special. I don't know that I can give a completely uniform answer, but it certainly seems reasonable.
For example in type A, modules for Cherednik algebras are realized using twisted D-modules on some stack. The parameter for twisting is an additive one, but the abstract category of twisted D-modules depends only on this parameter mod integral translations (hence translation functors). At integral points (or rather "the" integral points) many special things happen -- there are geometric obstructions to the existence of objects with various supports, and these obstructions vanish integrally -- which is why say category O for Lie algebras becomes much bigger integrally. More generally twisted D-modules can be described as sheaves on a gerbe, which depends only on the twist mod integers. If (and certainly only if) the parameter is rational - ie the twist is a torsion element - then you might expect to represent your gerbe by an Azumaya algebra, or equivalently to have finite rank twisted sheaves. I would imagine this general type of phenomenon is behind the results you mention, though I haven't thought about the specifics. But in any case this is a geometric phenomenon about categories of twisted D-modules in general, and as we know "basically all interesting categories of representations are some categories of twisted D-modules" so this is quite general.