Are these kept anywhere? I find myself occasionally amused by the usual forum stats (like total # of posts, average posts per day, most-visited forum, etc), and I think it would be cool if they were somewhere. Particularly with the integrated forums to the site, I personally think it'd be neat if at least post count or something were displayed on our profiles.

Not yet - adding stats pages is definitely something that's going to happen sooner rather than later. All the data is there, it just needs to be mined and displayed appropriately.