Indeed this is just the simplicial set associated to the clique simplicial complex of a graph. Any flag complex is (by definition) the clique complex of its 1-skeleton considered as graph. In particular the barycentric subdivision of any simplicial complex is a flag complex and consequently any simplicial complex is homeomorphic to a clique complex.

I think the idea is that (possibly with some stretch of interpretation) it is supposed to mean complete cut (tomic, as also in cyclotomic, meaning cut), i.e., complete intersection. I have also heard Messing attribute it to Mazur. As to why the answer should be obvious (as Clark pointed out), the "flat, locally of finite presentations and local complete intersection topology" is not something you would want to use more than once in a lecture (if that).

@David: Yes, the non-split case is a problem for the relative Tannakian which will no doubt result in me actually reading through your answer :-)

@BCnrd: It may very well be a question of style. Personally, I prefer to get an intuitive picture and worry about details when I have to (of course when I do have to it often enhances my intuition). Having étale toposes lying above the category of schemes is to me an intuitively satisfying picture irrespective of whether or not I throw in universes. If I actually want to go beyond intuition I know that I have to throw in universes. This makes the distance between intuition and mathematical reality shorter than if I have to worry about bounding everything so as to avoid universes.

This argument goes back to Grothendieck. The only thing that Grothendieck didn't have, which makes a proof of the formula much more natural, is the formula for the degree of a line bundle on a DM-stack in terms of the zero set counted as one does on stacks.

I would say it is very much like the axiom of choice; universes can mostly be avoided (even more so than in the case of AC) but it would force one to spend time on the least interesting parts of the theory and results would have to be qualified so as to make them more difficult to understand. I doubt very much that there are interesting results which would not allow such a reformulation. It is however very convenient when one considers things like the fibered category of étale toposes over the category of schemes.

... In this case the group scheme is just $K$ with the given action of $H$ (of course it may not be a finite group scheme so its affine algebra may not lie in $Rep_k(H)$ but rather in the category of ind-objects for $Rep_k(H)$.

A more Tannakian answer along the same lines is relative Tannakian theory. Restriction to $H$ gives a fibre functor $Rep_k(G) \to Rep_k(H)$ and the surjection $G\to H$ gives a "constant object" functor $Rep_k(H) \to Rep_k(G)$. One can imitate the usual Tannaka theory replacing $Rep_k(1)=Vec_k$ by $Rep_k(H)$ getting a description of $Rep_k(G)$ as the category of modules in $Rep_k(H)$ of some group scheme in $Rep_k(H)$. ....

See amendment to the answer.

Yes, to the first question. For the second part I am not suggesting that my example says something about direct images of affine morphisms. It is rather an examples of the local picture on the base (in this case $J$) while the direct image is along a proper map.

Unfortunately I made a stupid thinko and the picture is not quite as nice, see corrected answer.