Richardson varieties over finite fields

The intersections of opposite Schubert cells have a very nice decomposition into products of tori and affine spaces due to Deodhar which, of course, induces such a decomposition of the Richardson. This decomposition is defined over $\mathbb{Z}$ (actually it works in any building), so it lets you count points, and the strata are combinatorially described by special subwords of a reduced decomposition of one of the words. I recommend reading the paper of Marsh and Rietsch.

This is both well known and complicated. The number of points on a Richardson variety over $\mathbb{F}\_q$ is given by the R-polynomials of Kazhdan and Lusztig. These are not the more famous Kazhdan-Lusztig polynomials, but they are related and are introduced in the same paper. There is a simple recursion for R-polynomials (see the Wikipedia link). My impression (but I am not an expert) is that experts do not think there will ever be a simple description like the rank generating function of some simple poset. It is true that, in many small cases, the description you give as the rank generating function of an interval is correct. I believe that there should be a geometric description of when this happens (maybe if the Richardson is smooth?) and also a pattern avoidance description, but I don't know the details.

Regarding the questions about cell decompositions: The obvious decomposition of a Richardson, that is to say, as intersections of Schubert cells with opposite Schubert cells, is a stratification. This means that the closure of each piece is a union of such pieces. The pieces are not $\mathbb{A}^n$'s. I don't believe Richardson's can be stratified by $\mathbb{A}^n$'s, but I'm not sure.

A lot of this probably does simplify when one of your Schubert's is the big cell, but I don't know exactly how.

There are a lot of gaps in this answer, so I'm making it community wiki in the hope that experts will come fill in the details. If not, I've at least given you a starting point.