Actually, it's likely the hangar. When you have issues of different shifting behavior from top to bottom gears, and especially those weird 'double-click' required, that's a strong sign of hangar misadjustment.

All the barrel adjuster stuff and cable smoothness are for small changes that will not cause these sorts of problems. The cable tension also should not affect one half of the cogset more than the other to the degree the OP is getting. Cable tension issues should NEVER give double-shift problems - I know this as I've tried both very loose and overly tight cable tension for experimentation, and the main thing they do is give shifting limited to the upper or lower portions of the cassette (you'll be missing gears) but it otherwise shifts 100% perfectly between those gears if your hangar is aligned.

I literally had the exact problems as the OP with double-shifting required as well and top-to-bottom errant shift behavior that was 100% hangar alignment. Unfortunately, I too messed with nearly everything else before getting to it (man I futzed with new cables and cable tension for so long!), despite knowing full well that you really have to start with hangar adjustment, and everything else takes care of itself.