Arkham Origins: Are achievements dead?

They told to shut down multiplayer december 4th, but actually, to this day, it still is up and running. You can still play online on PC (maybe they only shut down online for consoles?).

No, but on window mode, because of this 4:3 monitor.

As I know, the season pass contains Cold, cold Heart, which contains more campaign missions with Mr. Freeze's wife, Initiation, with pre-Batman missions, and costumes, that you aren't allowed to use until finish the story, and that sucks. But for $3 is a good deal, I guess, since I didn't got it yet.

totally, and it's a good point to start enjoying :)
But I highly recommend you to follow the release order: Asylum>City>Origins>Blackgate
I know that Origins is a prequel, but it's all about gameplay evolution

Yes, Arkham games can be enjoyable for totally new people to DC comics. While they cater a lot to fans (especially the first two games, where one of the more famous Batman writers was greatly involved), they tell their own story in a fully standalone world.

Yes. I'm not into Batman or comics at all, but I absolutely loved Arkham Asylum (and Arkham City to a slightly lesser extent). I think any fan of action-adventure will enjoy these games.