I think when $F$ is finite you mean Deligne--Lusztig theory (although Kazhdan--Lusztig theory also plays a role). For $\mathrm{GL}_n(q)$ there is also the formidable work of Sandy Green ("The characters of the finite general linear groups", 1955, Trans. Amer. Math. Soc.).

If $x$ is semisimple and $F$ is algebraically closed or a finite field then one can appeal to the theory of reductive algebraic groups to obtain the centralisers. In particular, $C_G(x)$ will be a Levi subgroup. So one simply needs to compute maximal Levi subgroups in $G$ (which can be determined by the root system). You may find Bonnaf\'e's paper "Quasi-Isolated Elements in Reductive Groups" helpful.

Well, things are slightly different between the unipotent and reductive cases. In the case of $U_n(q)$, according to Boyarchenko, the characteristic functions of character sheaves are (up to multiplication by a power of $q$) irreducible characters. This is not true in the reductive case. In the unipotent case one only has a labelling by objects in the bounded derived category of constructible sheaves. It is a miracle that in the reductive case you can label the characters (and character sheaves) by objects living in the algebraic group. This seems not (or not known) to be the case in $U_n(q)$.

This is true. I somehow didn't make it clear how starkly different the situations are. The point I wanted to make was that the wildness is somehow contained in parameterising the families $\mathcal{F}_i$. If you could parameterise these families depending on $q$ then this would probably turn out to be a tame problem, as in your example. I suppose the point to make is that you have no hope of parameterising the families $\mathcal{F}_i$ even though the families themselves are a nice way to bunch up the characters.

I'm not sure about all Coxeter groups but certainly for finite real reflection groups there exist combinatorial descriptions of the classes and characters. See the book "Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras" by Geck and Pfeiffer (2000) for (many) more details.

So, if I've understood what you're saying correctly, $\Gamma_{n,m}$ is isomorphic to $(\mathbb{F}_p)^m \times \mathrm{GL}_{n-m}(p)$? In which case you're looking at a Levi subgroup of $\mathrm{GL}_n(p)$. Inducing from such subgroups is quite intractable on the whole. However if you first lifted the representation to a parabolic subgroup containing $\Gamma_{n,m}$ and then induce, then there is a very nice answer to your problem.

As far as I am aware, no. I have never seen any interaction between the 2-sided cells of an affine Weyl group and the character theory of finite reductive groups. Certainly in the framework of unipotent supports that I am talking about above, this definitely does not play any role.