What are 3D games with 2D sprites like Doom called

There's not really a "genre label" for this kind of gaming, simply because non-interactive games aren't much of a game genre. Most of the ones that exist are either silly idling games (Idle RPG/Progress Quest/Idle 2), bots (like this one that plays a roguelike by itself), or more AI experiment than game (Conway's Game of Life, for instance).

I think that "Zero Player Games" or "bot game" is about the best you can do. I did find some interesting articles when I searched for "games that play themselves."

There's also a subgenre of the strategy genre where you'll find games where you can only interact indirectly with the game world, with games like Populous or Evil Genius. This subgenre is generally called "Artificial Life" or "God Game".

Some games are capable of playing themselves, in a demo or "attract" mode. Arcade games are especially fond of this, as it attracts people to the cabinet and shows them what they might get for dropping in a quarter.

If we consider things like ProgressQuest games, then screensavers might be considered non-interactive games as well (or in reverse, non-interactive games might be screensavers). There was one screensaver for Win95 that played a little maze, for instance. Of course, there's also the iconic After Dark package that not only featured a number of things you might consider to be non-interactive games, but also some screensavers that were fully playable, like Lunatic Fringe.

Part of the problem is that we already have many non-interactive forms of digital entertainment. We give them many terms, but one is "movie." You could consider a movie like Scott Pilgrim vs. The World to be a "game that plays itself." The protagonist even levels up, has extra lives, and fights a final boss. Or, take the movie Clue, which went so far as to include multiple endings, and each time you went to see it there was a chance you'd see a different one. (I realize that's tangential, but it's sort of an interesting aside to your question.)

"Non Games" is a similar term, but it's more for games that have no goal, like Mario Paint. These games ride the line between game and application.

The question is hard to answer, mostly because the notion of supporting is vague. There is no clear definition of it when it comes to the game itself. Also, you cannot apply the terminology of another type of game. Since their is now particular definition for the game, I can only respond to the question based on the dictionary.

Dictionary will say that supporting is "to act in a secondary or subordinate role to (a leading performer)". With this definition, Annie, or Zyra can be used to support an AD Carry. Their main function is not to provide a burst of damage, like they would do as midlaners, but to be able to reliably control the enemies by landing stuns or root.

Of course they will build more like mages in the end game, mostly because their kit benefits from AP more than other stats. But traditional supports tends to do as well now: Lulu shield, speed boost or slow also scales on AP.

Furthermore, itemisation of such supports will strongly vary from a mage build. They will more probably go for utility first, with Sightstone for Warding and Cooldown Reduction to ensure they can get more stuns and root.

As a conclusion, I think that supporting is not depending on champion (even if most champion would be weak supports). Support in League of Legend is a role, it is not a category of champion. For being a support, you need to be able to provide for your carry. That can be : crowd control, sustain, protection or maybe other things. In my opinion, the meta has not changed yet. Gamestyle, however, has.