I remember Wendell saying that a 40 inch 4k monitor is the same width(and really close pixel density) as a 3440x1440 at 34 inches. So my question is can I run games with black bars at the 21:9 3440x1440 for games and still keep the 40 inch real state for productivity
I've not tried but I understand that, generally speaking, yes. It's possible that a specific monitor wouldn't behave well so verify before you buy one but that idea is pretty solid.
Yes you can. It's not that convenient to set up but you definitely can
I played around with an AMH A309U for a few months and the compromise for gaming was to run a borderless window on a black desktop, like Witcher 3 below, or create a custom 3440x1440 resolution and run the monitor in 1:1 pixel mode. Your monitor scaler will have some affect on your options as some games don't run so well in a borderless window. Dying light for example worked better 16:9 1920x1080 full screen.