You are asking a number of related questions here, most of which require more reading of Lusztig's papers. See the reference list in my conference paper here, for example. But note first that the notions about cells you mention are defined quite generally for Coxeter groups in the landmark 1979 paper by Kazhdan and Lusztig here. What is special about affine or finite Coxeter groups is the precise detail Lusztig later worked out. For example, in the fourth part of his series of papers on cells in affine Weyl groups, he arrived at a proof of his conjectured bijection between 2-sided cells and nilpotent orbits in the Lie algebra of a Langlands dual algebraic group of the relevant type. [But it is very hard to get an intuitive feeling for this bijection. As far as I know, there is not yet an alternative to Lusztig's complicated proof, which uses virtually all possible tools. So this part of the subject remains challenging.]
There is still no proof of an older 1983 conjecture by Lusztig on the precise number of left cells in a 2-sided cell. This and other such combinatorial matters are related to unsolved problems in representation theory. Anyway, the precise relationship between cells in affine Weyl groups and finite Weyl groups has been worked out in Lusztig's papers and is nicely illustrated in the rank 2 cases in his Japanese conference paper. (See also the photo of Lusztig in his specially made $G_2$ t-shirt in the paper by Gunnells here.)
Concerning canonical left cells, there has been further computational work by Chmutova and Ostrik in a 2002 paper here. Gunnells and others have gone on to study further the difficult problem of characterizing the location of "distinguished involutions" in all left cells, which I suspect is closely related to modular representations of the corresponding Lie algebras.
ADDED: To comment a little further, the occurrence here of the Langlands dual group is perhaps mysterioous, but for the reflection groups and cells it shows up mainly in the interchange of Lie types $B_\ell$ and $C_\ell$. The embedding of a Weyl group into an affine Weyl group is less affeccted, since Weyl groups of these types are isomorphic. But the dual version of an affine Weyl group, with its translation lattice expanded by a prime factor $p$, is crucial in characteristic $p$ representation theory of the algebraic groups involved (as Verma first noticed).
From Lusztig's work (especially his definition of "special" nilpotent orbits), one sees that a two-sided cell of the Langlands dual affine Weyl group meets a (unique) two-sided cell of the finite Weyl group precisely when the corresponding nilpotent orbit is special. In type $A_\ell$ where $W = S_{\ell+1}$, all orbits are special, but otherwise not. This adds a further layer of interest to the kind of parametrization you ask for. Special orbits have a Lusztig-Spaltenstein duality, for type $A_\ell$ just the transpose involution on partitions of $\ell+1$. This leads further, to Lusztig's special pieces of the nilpotent variety, which people now understand better in terms of an "exotic" version of the variety: see for example here.
Some short unpublished notes (in a more general context, where the finite Weyl group appears as one of the parabolic subgroups of an affine Weyl group) are here, but don't deal with the geometric side.
