I'll give a bunch of my tradelist for Doom Beta key.

Oh, I get fandom, and despite its publisher and my general aversion to the FPS franchise, I probably will play Doom 4 (I'm among the few who liked Doom 3 back then). I just have a hard time to imagine waiting for a Doom game to play it in multiplayer-only. I mean Doom 3 was known for a few things (insane HW reqs, simple mechanics, reliance on jump scares), but enjoyable MP was not of them. (Heck, the thing I distinctly remember reading about it was how much flak the MP part got in the reviews, even among those that tried to praise the questionable SP decisions.)

I know, this is why I said it stopped at that two. But when was the last time you saw anybody playing Doom 3 MP? Or heck, any Doom MP? :)
If it really switches to MP focus, despite its SP origins and letting Quake be the MP game, then I guess I can chalk it up to Bethesda really having zero knowledge in their own IPs. (Although after what the company did to Fallout – or, well, their own world –, I guess this was coming anyway.)

That is the normal procedure, yes. Also, if the MP part had some sort of levelling/unlocking system, it gets reset or the user account deleted. Beta is more like learning the mechanics early if you plan to stick around (and hoping players will do as well).

Also, I think we also heard Titanfall's MP (which was practically the entire game anyway) was "awesome" as well, and we all know how it turned out in the end. I cannot really believe any marketing before the game is out or at least the beta can be streamed. (When the beta cannot be even streamed, this is where you can be suspicious that the publisher knows its product is anything but enjoyable. Again, just look at Overwatch: closed beta, yet Twitch has constant coverage on it.)