Can I set the cover I want displayed for each grouping?
No (or not out of the box, maybe their is some expression). Album covers are random, but does change on mouse hover
Yes, I have something other than those displayed ([Name]), but when I change it it's no different: random album cover from among the albums grouped there. How random is this? It doesn't seem to be changing with every re-fresh/new login. But I can't tell how the cover is chosen.
just to make sure I'm on the same page, if you group by Artist (or Album Artist or AlbumArtist (auto)) you should get soemthing like this -- the [Name] field is the track name, don't think you want that there. On drill down over Beastie Boys here you would see something like this?
If your are just talking about eye-candy though It would be possible to manually stack albums, but if you have a big library can't see how this would be viable. Or you could make one using a 3D view.